Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory

To what extent does Noether's principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law, and show that this relates to physically impossible transformations in quantum theory, such as time-reversal and spin-inversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group~$G$ becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel. In particular, we investigate in detail the $U(1)$ and $SU(2)$ symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin-$j_A$ system can be used to polarise a larger spin-$j_B$ system, and on how much one can invert spin polarisation using a rotationally-symmetric operation. Finally, we also establish novel links between unitarity, complementary channels and purity that are of independent interest.


I. INTRODUCTION A. Symmetry principles versus conservation laws
Noether's theorem in classical mechanics states that for every continuous symmetry of a system there is an associated conserved charge [1][2][3].This fundamental result forms the bedrock for a wide range of applications and insights for theoretical physics in both nonrelativistic and relativistic settings.Quantum theory incorporates Noether's principle at a fundamental level, where for unitary dynamics generated by a Hamiltonian H we have that an observable A is conserved, in the sense of ψ|A|ψ being constant under the dynamics for any state |ψ , if and only if [A, H] = 0.In quantum field theory, Noether's theorem gets recast as the Ward-Takahashi identity [4,5] for n-point correlations in momentum space.
In all of the above cases a continuous symmetry principle is identified with some conserved quantity.However, the most general kind of evolution of a quantum state, for relativistic or non-relativistic quantum theory, is not unitary dynamics but instead a quantum channel.This broader formalism includes both unitary evolution and open system dynamics, but also allows more general quantum operations such as state preparation or discarding of subsystems.It is therefore natural to ask about the status of Noether's principle for those quantum channels that obey a symmetry principle.
A quantum channel E [6] takes a quantum state ρ A of a system A into some other valid quantum state σ B = E(ρ A ) of a potentially different system B. The channel respects a symmetry, described by a group G, if we have that for all g ∈ G, where U (g) denotes a unitary representation of the group G on the appropriate quantum system.However, even in the simple case of the U(1) phase group U (θ) = e iθN generated by the number operator N , we know from quantum information analysis in asymmetry theory [7], that situations arise in which the symmetry constraint is not captured by N := tr(N ρ) being constant [8].Indeed, even if we were given all the moments N k of the generator N of the symmetry, together with all the spectral data of the state ρ A , this turns out to still be insufficient to determine whether ρ A may be transformed to some other state σ B while respecting the symmetry.Conversely, given a symmetry principle, there exist quantum channels that can change the expectation of the generators of the symmetry in non-trivial ways.These facts imply that a complex disconnect occurs between symmetries of a system and traditional conservation laws when we extend the analysis to open dynamics described by quantum channels, see Fig. 1.Given this break-down of Noether's principle, our primary aim in this work is to address the following fundamental question:

Charge conserving operations
< l a t e x i t s h a 1 _ b a s e 6 4 = " j K z D u f 1 n Z d N O f O J M P m C B + 0 7 Q L 1 4 = " > A A A C E X i c b V C 7 T s M w F H X K q 4 R X g Z H F o k L q V C V d Y K z o w l g k + p C a q H K c 2 9 a q Y 0 e 2 U 6 m q + g s s / A o L A w i x s r H x N 7 h p B 2 g 5 k q W j c 8 6 1 f U + U c q a N 5 3 0 7 h a 3 t n d 2 9  r + + 8 i Z p 1 6 q + V / X v a + X 6 7 a q O I r p A l 6 i C f H S N 6 u g O N V E L U f S I n t E r e n O e n B f n 3 f l Y R g v O a u Y c / Y H z + Q P j l p 2 s < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j K z D u f 1 n Z d N O f O J M P m C B + 0 7 Q L 1 4 = " > A A A C E X i c b V C 7 T s M w F H X K q 4 R X g Z H F o k L q V C V d Y K z o w l g k + p C a q H K c 2 9 a q Y 0 e 2 U 6 m q + g s s / A o L A w i x s r H x N 7 h p B 2 g 5 k q W j c 8 6 1 f U + U c q a N 5 3 0 7 h a 3 t n d 2 9  r + + 8 i Z p 1 6 q + V / X v a + X 6 7 a q O I r p A l 6 i C f H S N 6 u g O N V E L U f S I n t E r e n O e n B f n 3 f l Y R g v O a u Y c / Y H z + Q P j l p 2 s < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j K z D u f 1 n Z d N O f O J M P m C B + 0 7 Q L 1 4 = " > A A A C E X i c b V C 7 T s M w F H X K q 4 R X g Z H F o k L q V C V d Y K z o w l g k + p C a q H K c 2 9 a q Y 0 e 2 U 6 m q + g s s / A o L A w i x s r H x N 7 h p B 2 g 5 k q W j c 8 6 1 f U + U c q a N 5 3 0 7 h a 3 t n d 2 9  r + + 8 i Z p 1 6 q + V / X v a + X 6 7 a q O I r p A l 6 i C f H S N 6 u g O N V E L U f S I n t E r e n O e n B f n 3 f l Y R g v O a u Y c / Y H z + Q P j l p 2 s < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j K z D u f 1 n Z d N O f O J M P m C B + 0 7 Q L 1 4 = " > A A A C E X i c b V C 7 T s M w F H X K q 4 R X g Z H F o k L q V C V d Y K z o w l g k + p C a q H K c 2 9 a q Y 0 e 2 U 6 m q + g s s / A o L A w i x s r H x N 7 h p B 2 g 5 k q W j c 8 6 1 f U + U c q a N 5 3 0 7 h a 3 t n d 2 9   Q1.What is the maximal disconnect between symmetry principles and conservation laws for quantum channels?Surprisingly, we shall see that this question relates to the distinction between the notion of an active transformation and a passive transformation of a quantum system.
B. Active versus passive: forbidden transformations in quantum mechanics.
In quantum mechanics the time-reversal transformation t → −t is a stark example of a symmetry transformation that does not correspond to any physical transformation that could be performed on a quantum system A [9].More precisely, within quantum theory timereversal must be represented by an anti-unitary operator Θ, and so cannot be generated by any kind of dynamics acting on a quantum system.Instead, time-reversal is a passive transformation -namely a change in our description of the physical system.On the other hand, active transformations, such as rotations or translations, are physical transformations with respect to a fixed description (coordinate system) that can be performed on the quantum system A. Time-reversal, therefore, constitutes an example of a passive transformation that is without any corresponding active realisation.This is in contrast to spatial rotations of A which admit either passive or active realisations.
If A is a simple spin system, then the action of time-reversal on the spin angular momentum J degree of freedom coincides with spin-inversion, which transforms states of the system as ρ A → T (ρ A ) = Θρ A Θ † .In the Heisenberg picture this transformation sends J → −J.Indeed, while spin-inversion is seemingly less abstract than time-reversal, it constitutes another symmetry transformation in quantum theory that is forbidden in general -a passive transformation with no active counterpart.
The strength of this prohibition on spin-inversion actually depends on the fundamental structure of quantum theory itself.This can be seen if we ask the question: what is the best approximation to spin-inversion that can be realised within quantum theory through an active transformation, given by a quantum channel E, of an arbitrary state ρ A to some new state E(ρ A )? If we restrict to the simplest possible scenario of A being a spin-1/2 particle system, we have that spin-inversion coincides with the universal-NOT gate for a qubit.It is well known that such a gate is impossible in quantum theory [10], and the best approximation of such a gate is a channel S − that transforms any state ρ with spin polarisation P(ρ A ) := tr(Jρ A ) into a quantum state S − (ρ A ) such that We refer to S − as the optimal inversion channel for the system.
It is important to emphasise that the pre-factor of −1/3 is fundamental and cannot be improved on.Its numerical value can be determined by considering the application of quantum operations to one half of a maximally entangled quantum state -anything closer to perfect spin-inversion would generate negative probabilities, and would thus be unphysical.Indeed, if we removed entanglement from quantum theory, by restricting to separable quantum states, then there would be no prohibition on spin-inversion on the system! 1hile this limit is easily determined for spin-1/2 systems, it raises the more general question: Q2.What are the limits imposed by quantum theory on approximate spin-inversion and other such inactive symmetries?
Here, an inactive symmetry transformation simply means a symmetry transformation that is purely passive and does not have an active counterpart.More precisely, and focusing in on spin-inversion, the question becomes: given any quantum system A, what is the quantum channel E that optimally approximates spin-inversion on A? For a d = 2 qubit spin system, this analysis essentially coincides with looking at depolarizing channels.However, for a d > 2 spin system, this connection with depolarizing channels no longer holds and a more detailed analysis is required to account for the spin angular momentum of the quantum system.
Spin -j A system < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 M u T

Spin inversion
< l a t e x i t s h a 1 _ b a s e 6 4 = " Y q k z m W L q Y h k i e d W 6 i c f t l W U H U s g = " > A A A B 9 X i c b V C 7 T s M w F L 0 p r 1 J e B U Y W i w q J q U q 6 w F j B w l g E f U h t q B z 3 p r X q O J H f X i W t W t V z q 9 5 t r V K / y u s o w g m c w j l 4 c A F 1 u I E G N I G B g m d 4 h T f n y X l x 3 p 2 P R b T g 5 D P H 8 A f O 5 w + s K p K b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Y q k z m W L q Y h k i e d W 6 i c f t l W U H U s g = " > A A A B 9 X i c b V C 7 T s M w F L 0 p r 1 J e B U Y W i w q J q U q 6 w F j B w l g E f U h t q B z 3 p r X q O J H t g K q o / 8 H C A E K s / A s b f 4 P b Z o C W I 1 k 6 O u d c 3 e s T J I J r 4 7 r f T m F t f W N z q 7 h d 2 t n d 2 z 8 o H x 6 1 d J w q h k 0 f X i W t W t V z q 9 5 t r V K / y u s o w g m c w j l 4 c A F 1 u I E G N I G B g m d 4 h T f n y X l x 3 p 2 P R b T g 5 D P H 8 A f O 5 w + s K p K b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Y q k z m W L q Y h k i e d W 6 i c f t l W U H U s g = " > A A A B 9 X i c b V C 7 T s M w F L 0 p r 1 J e B U Y W i w q J q U q 6 w F j B w l g E f U h t q B z 3 p r X q O J H t g K q o / 8 H C A E K s / A s b f 4 P b Z o C W I 1 k 6 O u d c 3 e s T J I J r 4 7 r f T m F t f W N z q 7 h d 2 t n d 2 z 8 o H x 6 1 d J w q h k 0 b n t v w 7 i 5 q z e s y j g o 4 A s e g D j x w C Z r g F r R A G y D w C J 7 B K 3 i z n q w X 6 9 3 6 m L U u W e X M A f g D 6 / M H 5 H u X q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j J a b l 6 b n t v w 7 i 5 q z e s y j g o 4 A s e g D j x w C Z r g F r R A G y D w C J 7 B K 3 i z n q w X 6 9 3 6 m L U u W e X M A f g D 6 / M H 5 H u X q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j J a b l 6 b n t v w 7 i 5 q z e s y j g o 4 A s e g D j x w C Z r g F r R A G y D w C J 7 B K 3 i z n q w X 6 9 3 6 m L U u W e X M A f g D 6 / M H 5 H u X q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j J a b l 6 b n t v w 7 i 5 q z e s y j g o 4 A s e g D j x w C Z r g F r R A G y D w C J 7 B K 3 i z n q w X 6 9 3 6 m L U u W e X M A f g D 6 / M H 5 H u X q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e q l j V 0 S P g o r l 6 q b 0 b l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e q l j V 0 S P g o r l 6 q b 0 b l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e q l j V 0 S P g o r l 6 q b 0 b l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e q l j V 0 S P g o r l 6 q b 0 b

FIG. 2. Spin-inversion and amplification.
There exist quantum channels that can invert or amplify the polarisation of a spin system while exactly respecting SU (2) rotational symmetry.The values κ± provide the ultimate limits of such processes and depend only on the dimension of the spin systems involved.

C. Structure and scope of the problem
In this paper, our main focus will be on the maximal disconnects between symmetry principles and conservation laws.We will focus on symmetries corresponding to Lie groups, and the dominant case will be the SU (2) rotational group.This provides an illustration of the non-trivial structures involved, but also shows that the problem of performing an optimal approximation to spininversion arises naturally.We do not consider more general inactive symmetries, but leave this to future work.
We first fully solve Q2 for the case of spin-inversion, and show that this can be better and better approximated at a state level as we increase the dimension of the spin.However, this has an information-theoretic caveat that things look quite differently at a quantum channel level.The solution of spin-inversion also connects with a seemingly paradoxical ability to perform spinamplification under rotationally symmetric channels.We diagrammatically present these results in Fig. 2.
Both spin-inversion and spin-amplification turn out to be two extremal deviations from Noether's principle, and thus lead on to the central question Q1.Here, we derive general bounds on deviations from conservation laws for general groups and systems.These describe the tradeoff between allowed deviations and the departure from closed unitary dynamics as schematically portrayed in Fig. 3.
The nature of the considered questions requires one to understand the structural aspects of the set of symmetric quantum channels and, in particular, to have a strong handle on the extremal points of this set.One also needs an operationally sensible way to cast questions Q1 and Q2 into quantitative and well-defined forms.To these ends we extend previous results on the structure of symmetric channels [11][12][13][14][15] and derive novel relations for the unitarity of a quantum channel [16] -both of which are of independent interest to the quantum information com-

munity.
Since our results provide general bounds on the behaviour of expectation values of observables under symmetric dynamics, we believe that they may be of relevance to scientists working in quantum open systems, decoherence theory, and quantum technologies [17].Moreover, as Noether's principle is fundamental and farreaching, our studies are of potential interest to people investigating foundational topics and relativistic physics [18,19].
The structure of the paper is as follows.In the next section we give a detailed overview of our main results, and then in the rest of the paper we gradually introduce all the necessary ingredients that allow us to rigorously address the questions posed here and derive our results.In Sec.III we introduce the notation and provide preliminaries on covariant quantum channels.Next, in Sec.IV we define quantitative measures of the departure from conservation laws and from closed unitary dynamics.Section V contains the technical core of our paper with a detailed analysis of the convex structure of the set of symmetric channels.In Sec.VI we then use these mathematical tools to address the problem of spininversion and amplification, while in Sec.VII we derive trade-off relations between conservation laws and decoherence.Finally, Sec.VIII contains conclusions and outlook.

A. The optimal spin-inversion channel
We first address question Q2 by studying in detail the problem of approximate spin polarisation inversion for spin-j A system A with 2j A + 1-dimensional Hilbert space H A .The higher-dimensional spin angular momentum observables J A := (J x A , J y A , J z A ) along the three Cartesian coordinates generate rotations corresponding to elements g ∈ SU(2), which act on the system via the unitary representations U A (g) describing the underlying symmetry principle.A channel E : B(H A ) → B(H A ) is symmetric under rotations, or SU(2)-covariant, if it satisfies Eq. ( 1) for all states ρ A ∈ B(H A ) and g ∈ SU(2) (since the input and output systems are the same we have B = A).Now, rotational invariance ensures that the symmetric channel E acts on single spin systems isotropically.As a result, spin polarisation vector P (ρ A ) of an initial state ρ A is simply scaled by the action of E, i.e.,

P (E(ρ
for a single parameter f (E) that is independent on ρ A or the spatial direction.The question Q2 thus amounts to determining the symmetric quantum channel S − with coefficient f (S − ) that is as close as possible to −1 (which can only be achieved by the unphysical spin-inversion operation).
As the set of all symmetric channels is convex, this becomes a convex optimisation problem whose solution is attained on the boundary of the set.The convex structure of SU(2)-symmetric quantum channels on spin systems has been previously examined by Nuwairan in Ref. [13], where a characterisation of extremal channels is given.We review these results in Sec.V A and extend the analysis in terms of the Liouville and Jamio lkowski representations of channels (see Sec. III for details).This, in turn, allows us to directly compute the scaling factors f (E) for any symmetric channel.
The convex set of SU(2)-covariant quantum channels on a spin-j A system forms a polyhedron with 2j A +1 vertices, each corresponding to a CPTP map E L labelled by an integer L ∈ {0, . . ., 2j A }. Therefore, any such symmetric channel E is a convex combination of these extremal covariant channels: where {p L } 2j A L=0 forms a probability distribution.The following result gives the best physical approximation to spin-inversion, and is proved and generalised to different input and output systems in Theorem 9 of Sec.VI B.
Result 1.The optimal spin polarisation inversion channel is achieved by S − := E 2j A , the extremal point of SU(2)-covariant channels with the largest dimension 2j A + 1 of the environment required to implement it.It results in an inversion factor: This generalises the previous result on optimal approximations of universal-NOT under rotational symmetry, and determines a fundamental limit that quantum theory imposes on the specific task of (universally) inverting the spin of a quantum system.The higher the dimension of the system, the larger is the maximal spin-inversion factor.Specifically, the optimal channel S − in the limit j A → ∞ approaches f (S − ) → −1, which is the value obtained under the inactive spin-inversion transformation.However, this feature alone does not imply that the channel S − behaves more like spin-inversion as the dimension of the system increases.As shown previously [8], once one goes beyond unitary dynamics, the angular momentum observables do not provide a complete description of symmetry principles and information-theoretic aspects become crucial.
To explicitly quantify this aspect, in Sec.VI C we compare the fidelity between the output of an active symmetric channel versus the passive transformation of spininversion T .We restrict to input states ρ A within the convex hull of spin coherent states as these behave classically in the sense of saturating the Heisenberg bound.We find that the output fidelity is given by which is maximised whenever p 2j A = 1, i.e., whenever E coincides with the optimal spin-inversion channel S − .Notice that while f (S − ) approaches −1 as we increase j A , the fidelity only achieves F (S − (ρ A ), T (ρ A )) → 1/2 in the limit, with the highest bound occurring for j A = 1/2.In other words, the actions of the symmetric channel E and the passive transformation T on quantities beyond P(ρ A ) distinguish the two, and limit the fidelity at the state level.

B. Spin amplification
The simple structure of the extremal points of SU(2)covariant channels generalises to the situation where the input and output spaces correspond to different irreducible spin systems.We discuss all these aspects in Sec.V A, and extensions to general compact Lie groups in Sec.V B. The convex set of symmetric channels E : B(H A ) → B(H B ), where H A and H B are Hilbert spaces for spin-j A and spin-j B systems, forms a polyhedron now with 2 max(j A , j B ) + 1 extremal points.In this scenario, it also holds that the spin polarisation of any input state is scaled isotropically by a constant parameter f (E), which depends only on the particular symmetric channel E. While for j A = j B , it was always the case that f (E) ≤ 1, this no longer holds true for j B > j A , and the spin can be amplified under a symmetric open dynamics.The ultimate limits of this are derived in Theorem 10, and are summarised as follows.
Result 2. Let us denote by κ + = max E f (E), where the maximisation occurs over the convex set of SU(2)-covariant channels E : B(H A ) → B(H B ). Then the maximal spin-amplification factor κ + is given by: The above result may initially seem paradoxical: using purely rotationally invariant transformations on a quantum system, we are free to arbitrarily increase the expectation value of angular momentum.This provides a dramatic example of the disconnect between symmetry principle and conservation laws.This surprising spinamplification effect requires that the dynamics is not unitary, but is instead given by a quantum channel with nontrivial Kraus rank, and the intuitions we acquire while dealing with unitary evolution fail badly when we look at more general open quantum dynamics.
But where does this new angular momentum come from?Here, the ability to perform approximate spininversion comes in.Any symmetric quantum channel can be purified to a Stinespring dilation involving a symmetric unitary V and an environment E in a pure state |η E with zero angular momentum [20][21][22], where we have that AE and BC denote the two different ways of factoring the global system.Since angular momentum is exactly conserved across the joint system AE we see that we must have where Ẽ denotes the complementary channel to E obtained by tracing out B after the action of the global unitary V [6].We now see that spin-inversion and spinamplification are complementary to each other.Namely, given any spin-amplification for which f (E) > 1, Eq. ( 9) necessarily implies that the complementary channel must have f (E) < 0, and thus is a spin-inversion channel.Some of these features have been discussed previously from the perspective of asymmetry theory [23], and earlier in relation to optimal cloning and the universal-NOT gate [24].In particular, the complementary channel of the optimal spin polarisation inversion channel S − will be the maximal spin amplification S− : B(H A ) → B(H B ) between a spin j A system and a spin j B = 2j A system.This generalises to optimal spin polarisation inversion channels between spin systems of different dimensions.
From the perspective of asymmetry theory, every resource measure is monotonically non-increasing under symmetric channels, and thus the fact that polarisation can be increased implies that spin polarisation cannot be a proper measure of asymmetry [23].The polarisation may increase, but its ability to encode a spatial direction must become inherently noisier.This is also in agreement with the No-Stretching Theorem [25] for spin systems.Starting from Q2, we analysed to what degree a spininversion is possible within quantum theory.This led us to consider symmetric quantum channels and we found that both spin-inversion and spin-amplification are directly related and can be approximately performed under the symmetry constraint.These two examples are maximal disconnects between symmetric dynamics and conservation laws, and thus bring us to the broader issue of question Q1.
In order to address it properly, we first need to define measures quantifying the deviations from conservation laws and from unitary dynamics.We also generalise the discussion to symmetries described by an arbitrary compact Lie group G, and introduce quantitative measures for probing how much the conserved charges associated with symmetry generators, {J k A } n k=1 and {J k B } n k=1 , can fluctuate between initial and final states, ρ A and E(ρ A ), for a G-covariant channel E. To that end, in Sec.IV we introduce the notion of average total deviation from a conservation law, which we define as the average L 2 norm of the difference in expectation values between ψ = |ψ A ψ A | and E(ψ) of the generators.Explicitly: where the integration is with respect to the standard Haar measure on pure states.
To quantify how close a channel E is to a unitary dynamics we employ the notion of unitarity, first defined in Ref. [16].It is defined as the average output purity over all pure states with the identity component subtracted, i.e., and satisfies u(E) ≤ 1 with equality if and only if E is a unitary channel.Note that previously this was defined only for channels between the same input and output spaces but, as we explain in Sec.IV, the definition can be generalised.We also provide a simple characterisation of unitarity in terms of the complementary channel, describing the back-flow of information from the environment, and relate it to the conditional purity of the corresponding Jamio lkowski state.These results, which may be of independent interest, can be summarised as follows.
Result 3. Let u(E) be the unitarity of an arbitrary quantum channel E from input system A to output system B, then 1. (Purity representation) where γ A|B (ρ is the conditional purity of a bipartite state, and J (E) is the Jamio lkowski state of quantum channel E.

(Complementary channel representation)
where Ẽ is the complementary channel to E in any Stinespring dilation.

(Zero decoherence)
We have that u(E) = 1 if and only if E is an isometry channel.
Thus, unitarity can be understood both as a puritybased measure of correlations in the Jamio lkowski state, or alternatively as a trade-off between the output purities for the channel and its complement.This result is independent of symmetry-based questions and holds for arbitrary quantum channels.
When do conservation laws hold?For a unitary symmetric dynamics, the corresponding conservation laws will always hold, but generally this is no longer true for symmetric quantum channels.There will be situations, however, when the degrees of freedom that decohere through interactions with the environment have no effect on the expectation values of the generators.In Sec.VII B we give the most general form of such a covariant channel that is unital and for which conservation laws always hold.Such behaviour would require the presence of decoherence-free subspaces, so that the information is protected from leaking into the environment.It follows that conservation laws will hold for symmetric dynamics that protects the degrees of freedom associated with the symmetry generators from leaking the information into the environment.More precisely, suppose that {J k A } n k=1 generate a unitary representation U A acting on the Hilbert space H A that describes the quantum system.Any symmetric channel In this sense, conservation laws may be viewed as a form of information preserving structures [26].Consider also a simple example of a two-qubit system AA , where only A carries spin angular momentum, so the symmetry generators are J x A ⊗ I A , J y A ⊗ I A and J z A ⊗ I A .Any channel of the form E AA = I A ⊗ E A is symmetric, with I A the identity channel on system A and E A an arbitrary quantum channel on system A .Moreover, E satisfies ∆(E AA ) = 0, so that the associated conservation laws hold despite the fact that E AA can be arbitarily far from unitary dynamics.This example illustrates that probing conservation laws for a physical realisation of symmetric dynamics will not always be sufficient to decide whether there are decoherence effects present.
In other words, robustness of conservation laws does not occur for all types of systems.Nevertheless, there are regimes that guarantee robustness for conservation laws.In such cases, approximate conservation laws hold if and only if the dynamics is close to a unitary symmetric evolution.For example, whenever B(H A ) contains a single trivial subspace then there is no symmetric channel other than identity for which conservation laws hold (which is the case, e.g., when H A carries an irreducible representation of SU (2)).
What does it mean for conservation laws to be robust under decoherence?If for all channels E obeying a given symmetry principle, it holds that ∆(E) ≈ 0 if and only if u(E) ≈ 1, we say that the associated conservation laws are robust.This can be established by finding upper and lower bounds on the deviation ∆(E) that coincide when u(E) → 1.In Sec.VII A we show in Theorem 11 that for all types of symmetries described by connected compact Lie groups, one can find such an upper bound (and the result extends to different input and output systems).
Result 4. Given any connected compact Lie group, for a symmetric channel E approximating a symmetric unitary the associated conservation laws will hold approximately.In other words.there exists an upper bound on the deviation from conservation law in terms of unitarity: for some constant M > 0 independent of E.
In order to obtain lower bounds, however, additional assumptions are required.It is clear from the previous discussion that conservation laws can hold beyond unitary dynamics, and in those situations we cannot expect to obtain lower bounds on the deviation in terms of unitarity.However, there exist symmetries for which conservation laws only hold for symmetric unitary dynamics, and then robustness is achieved.This happens in the case of spin-j system with symmetry generators given by higher-dimensional spin angular momenta generating an irreducible representation of SU (2).We prove the following result in Theorem 13 of Sec.VII B.
Result 5.For a spin-j system, spin angular momentum conservation laws are robust to noise described by a symmetric channel E the following bounds hold: More generally, we prove in Theorem 12 that whenever the quantum system carries a representation U A of a Lie group G for which U A ⊗ U * A has a multiplicity-free decomposition, then the associated conservation laws are robust under any open system dynamics given by the symmetric channel Finally, in Sec.VII D we obtain specific upper bounds on the deviation from a conservation law for energy that generates a U(1) symmetry constraint, in terms of the unitarity of the U(1)-symmetric channel.We also explain why a lower bound cannot hold because of the many multiplicities that appear in the decomposition of B(H A ).This analysis relies on the structure of convex set of U(1)covariant channels, which we expand on in Sec.V C.

III. NOTATION AND PRELIMINARIES
A. Quantum channels and their representations A state of a finite-dimensional quantum system A is described by a density operator ρ A ∈ B(H A ), with B(H A ) denoting the space of bounded operators on a d A -dimensional Hilbert space H A , that also satisfies ρ A ≥ 0 and tr (ρ A ) = 1.The space B(H A ) is itself a Hilbert space with the Hilbert-Schmidt inner product X, Y = tr X † Y .General evolution between d Adimensional and d B -dimensional quantum systems is described by a quantum channel E given by a linear superoperator E : B(H A ) → B(H B ) that is completely positive and trace-preserving (CPTP).More broadly, we will also consider CP maps, i.e., linear superoperators that are only completely positive (CP), but not trace-preserving Closed dynamics is described by a unitary channel for any X ∈ B(H A ).It is then straightforward to show that the entries of L(E) are given by 18) Note that, in the Liouville representation, the composition of quantum channels becomes matrix multiplication, i.e., L(E • F) = L(E)L(F).
One can also represent a quantum channel E via its Jamio lkowski state J (E) ∈ B(H B ) ⊗ B(H A ) defined by where I A denotes the identity channel acting on B(H A ).
The condition for complete positivity of E is equivalent to the positivity of J (E), while the trace-preserving property of E correspond to tr B (J (E)) = I A /d A .We note that we may pass from the Liouville representation to the Jamio lkowski representation via where R is the reshuffling operation defined as the linear operation for which |ab cd| R = |ac bd| for all computational basis states.
Finally, any quantum channel E admits a Stinespring representation in terms of an isometry V : for all X ∈ B(H A ).The isometry V that defines the quantum channel E is unique up to a local isometry on the environment.Note that, using the above, the adjoint channel E † is given by for all Y ∈ B(H B ).
Stinespring representation allows one to introduce the concept of a complementary channel : a quantum channel Ẽ is complementary to E, defined by Eq. ( 21), if its action is given by We also note that the adjoint of the complementary channel, which we denote by Ẽ † , is given by for all X ∈ B(H E ).

B. Symmetries and G-covariant channels
Consider a group G that acts on H A and H B via unitary representations g → U A (g) and g → U B (g), so that the group action on quantum states is given by unitary channels Recall that every finitedimensional unitary representation on a Hilbert space is the direct sum of irreducible representations, or irreps.We say that a quantum system A is an irreducible system if H A carries an irrep of G, i.e., if H A has no non-trivial subspace closed under the action of U A (g).
We say that a quantum channel E : We will now explain how the above definition affects different representations of a covariant quantum channels.Before that, however, we need to state a crucial result, the proof of which can be found in every textbook on representation theory.
Lemma 1 (Schur's lemma).Let U (g) be an irreducible representation of a group G on a Hilbert space H.Then, any operator X ∈ B(H) satisfying [X, U (g)] = 0 for all g is a scalar multiple of identity on H.Moreover, if Let us start with the structure of the Liouville representation of G-covariant channels.
Theorem 2. Let U A (g) and U B (g) be the unitary representations of G on H A and H B .Then, the Liouville representation of a G-covariant channel where λ ranges over all irreps that appear in both irrep decompositions of tensor representations U A (g) ⊗ U * A (g) and U B (g) ⊗ U * B (g), I λ are the identity matrices acting within the irrep subspaces, and L λ denote non-trivial m λ B ×m λ A block matrices acting on the multiplicity spaces.
Proof.First, using the Liouville representation, the covariance condition is equivalent to Note that L(U g A ) = U A (g) ⊗ U * A (g) is itself a (tensor) representation of G, and an analogous statement holds for L(U g B ). Therefore, we can decompose them into irreps as where λ ranges over all irreps that appear in each decomposition, and the group acts trivially on the multiplicity spaces of dimensions m λ A and m λ B .Now, since the covariance condition means that L(E) commutes with group representations having the above decompositions, the Shur's lemma implies that L(E) acts non-trivially only on the multiplicity spaces, leading to the decomposition given in Eq. (26).
Next, let us proceed to the Jamio lkowski representation of a covariant channel E. Theorem 3. Let U A (g) and U B (g) be the unitary representations of G on H A and H B .
Then, the Jamio lkowski representation of a G-covariant channel E : B(H A ) → B(H B ) is given by where, λ ranges over all irreps that appear in the irrep decomposition of tensor representation U B (g) ⊗ U * A (g), I λ are the identity matrices acting within the irrep subspaces, and J λ denote non-trivial square matrices of size m λ BA × m λ BA that act on the multiplicity spaces.Proof.The covariance condition means that for all g ∈ G we have By employing the fact that for any unitary U we have Combining the above two equations we find that covariance of E is equivalent to J (E) satisfying the following commutation relation: As in the proof of Theorem 2, we can decompose the tensor representation appearing in the above commutator into irreps, Once again, by using the Schur's lemma, we arrive at the block-diagonal decomposition of J (E) given in Eq. (29).
Finally, there is also a very particular form of the Stinespring representation of a G-covariant channel given by the following theorem.
Theorem 4. Given a G-covariant channel E, there exists an environment system E, with a Hilbert space H E and a unitary representation U E (g), together with a Gcovariant isometry V : for all X ∈ B(H A ).
The proof of the above result can be found in Ref. [21].

C. Irreducible tensor operators
The set of operators {T λ,α k } λ,α,k in B(H A ) are called irreducible tensor operators (ITOs) if they transform irreducibly under the group action, where λ labels irreducible representations of G with matrix elements v λ kk , and α denotes multiplicities.From the above property it can be deduced via Schur's orthogonality theorem that the set of ITOs must be orthonormal, tr((T λ ,α k Throughout the paper we will denote the ITOs for the input system, living in B(H A ), by T λ,α k , and the ITOs for the output system, living in B(H B ), by S λ,α k .These yield symmetry-adapted bases for B(H A ) and B(H B ) that are particularly useful for the studies of Gcovariant channels.More precisely, by employing the block diagonal structure of the Liouville representation for such channels stated in Theorem 2, and using the defining property of ITOs, we have Moreover, since ITOs are orthonormal, any density matrix in B(H A ) (and analogously for B(H B )) can be written as where we denoted the vector of ITOs transforming under a λ-irrep by T λ,α = (T λ,α 1 , ..., T λ,α d λ ), with d λ being the dimension of λ-irrep.

D. Continuous symmetries and conserved charges
Continuous symmetries of the system A are related to compact Lie groups.The representation of such a group G can be generated by infinitesimal generators {J k A } n k=1 .For simply connected Lie groups, representations of the group are in a one-to-one correspondence with representations of the Lie algebra g via the exponentiation map.More precisely, we have with g k ∈ R continuously parametrizing the group action.In such a Lie algebraic setting, by considering infinitesimal group action, g k → 0, one can show that the covariance of a linear map E : B(H A ) → B(H B ), specified by Eq. (25), is equivalent to for all k ∈ {1, . . ., n} and X ∈ B(H A ), with [X, Y ] denoting a commutator.By taking the Liouville representation of the operators on both sides of the above equality and employing the identity |XY Z = X ⊗ Z * † |Y , one can alternatively express the covariance condition as for all k.In particular, for a unitary G-covariant channel V : B(H A ) → B(H A ), the condition becomes simply [V, J k A ] = 0.As a result, for all k and for all quantum states ρ A ∈ B(H A ) we have tr i.e., the generators of the symmetry, {J k A } n k=1 , give the conserved (Noether) charges.

IV. DEVIATIONS FROM CLOSED DYNAMICS AND FROM CONSERVATION LAWS
The main aim of this paper is to quantitatively investigate the deviation from conservation laws as the symmetric dynamics deviates from being closed.In order to achieve this, we obviously need to understand the structure of covariant quantum channels that model symmetric open dynamics, and we will pursue this task from Sec. V onwards.However, there is also one more crucial ingredient needed for our analysis: namely, we need quantitative measures of how much a given dynamics deviates from being closed, and how much it deviates from satisfying the conservation law.In this section we introduce such measures and provide their basic properties.

A. Quantifying the deviation from closed dynamics
In order to quantify how much the dynamics generated by a given quantum channel E deviates from the closed unitary dynamics we employ the notion of unitarity.It was originally introduced in Ref. [16] as a way to quantify how well a quantum channel preserves purity on average.We extend these results to allow for distinct input and output system dimensions for a quantum channel E : B(H A ) → B(H B ). Definition 5. Unitarity of a quantum channel E : B(H A ) → B(H B ) is defined as the average output purity with the identity component subtracted: the integral is taken over all pure states ψ = |ψ ψ| ∈ B(H A ) distributed according to the Haar measure.
As we prove in Appendix B, the above extension of unitarity satisfies the original condition u(E) ≤ 1 with equality if and only if the operation is an isometry (as opposed to a unitary in the original formulation).This means u(E) = 1 is equivalent to the existence of an isometry V : H A → H B such that E(ρ) = V ρV † .Furthermore, as shown by the authors of Ref. [16], unitarity can be efficiently estimated using a process similar to randomised benchmarking, and can be calculated using the Jamio lkowski representation of E. This characterisation through J (E) carries over to the extended version we discuss here and, moreover, we find a novel characterisation of u(E) in terms of the output purity of E and its complementary channel Ẽ.These results are summarised in the following lemma (see Appendix B for the proof).Lemma 6. Unitarity of a channel E : B(H A ) → B(H B ) can be equivalently expressed by the following relations: with γ(ρ) = tr(ρ 2 ) denoting the purity of a state ρ.
Finally, let us remark that Eq. ( 44) suggests defining the notion of conditional purity for a bipartite system, Then, unitarity of a channel is simply expressed by the scaled conditional purity of its Jamio lkowski state:

B. Quantifying the deviation from conservation laws
Typically, the expectation values of symmetry generators, {J k } n k=1 , are not constant under non-unitary Gcovariant dynamics.In order to quantify this deviation from conservation laws we need to introduce appropriate measures.For any quantum operation E we define the directional deviation ∆ k for the expectation value of the J k generator with respect to the state ρ A as Note that by introducing with E † denoting the adjoint of E that describes its action in the Heisenberg picture, we can rewrite Eq. (48) as As we are equally interested in the deviation from a conservation law for all conserved charges, we define the total deviation ∆ tot as the l 2 norm of directional deviations for all generators: Finally, since we aim at quantifying how much a channel deviates from conservation law, independently of the input state, we introduce the average total deviation ∆(E): where we integrate with respect to the induced Haar measure over all pure states ψ ∈ H A .The above expression for the average total deviation ∆ can clearly be rewritten in the following form Next, we can employ the identity where π is any permutation on N symbols and P π is the corresponding Hilbert space unitary.In our case N = 2, so we only have the identity I and the flip unitary operation F. Thus,

V. CONVEX STRUCTURE OF SYMMETRIC CHANNELS
We now proceed to investigate the convex structure of the set of symmetric channels E : B(H A ) → B(H B ), with a particular focus on its extremal points.We start with a specific example of SU(2)-covariant channels, the convex structure of which was investigated before in Ref. [13].In this case, we provide a full characterisation of the extremal symmetric channels between irreducible systems, i.e., with Hilbert spaces of the input and output systems, H A and H B , corresponding to spin-j A and spin-j B systems with d A = 2j A + 1 and d B = 2j B + 1.The technical results derived here will be then employed in Sec.VI to study optimal covariant channels for spin-inversion and spin amplification.Next, we switch to a generic case of a compact group G. Here, we describe a useful decomposition of symmetric channels, which will be crucial in Sec.VII to analyse the trade-off between deviations from conservations laws and deviations from closed symmetric dynamics.We also explain how, under the assumption of multiplicity-free decomposition, this leads to a complete characterisation of the extremal points of G-covariant channels: the corresponding Jamio lkowski states are then given by normalised projectors onto irreducible subspaces.Finally, we investigate the U(1) group, which is the extreme example of a group that does not have a multiplicity-free decomposition (i.e., since U(1)-irreps are one-dimensional, all the non-trivial dynamics happens within the multiplicity spaces).In this particular case, which is physically relevant due to its connection with conservation law for energy, we find an incomplete set of extremal channels, which is however large enough to generate arbitrary action on the multiplicities of the trivial irrep λ = 0 (which physically encodes the action of the channel on energy eigenstates).

A. Extremal SU(2)-covariant channels between irreducible systems
The Lie group related to rotations of a system A in physical three-dimensional space is the SU(2) group.It has three generators, {J x A , J y A , J z A }, corresponding to angular momentum operators along three perpendicular axes, which generate general rotations.The unitary representation of such a rotation on the Hilbert space H A is given by with g k ∈ [0, 2π] parametrising the rotation angles.Irreducible representations of SU(2) group can be classified according to total angular momentum j, which is either an integer or half-integer.The j A -irrep is (2j A + 1)-dimensional and the corresponding subspace of H A is spanned by {|j A , m } j A m=−j A , which are the simultaneous eigenstates of total angular momentum, J 2 A = (J x A ) 2 + (J y A ) 2 + (J z A ) 2 , and J z A , with eigenvalues j A and m, respectively.Here, we focus on systems whose Hilbert space H A carries a j A -irrep, i.e., H A is spanned by d A = 2j A + 1 vectors |j A , m that transform as the j A -irrep (also meaning that there is no subspace of H A that is left invariant under the action of U A (g)).Physically this corresponds to a simple spin-j A system rather than to the one composed of many spin-j systems.
The set of SU(2)-covariant channels between a system whose Hilbert space carries j A -irrep and a system whose Hilbert space carries j B -irrep has a particularly simple structure.This is because the representations U A (g) ⊗ U * A (g) and U B (g) ⊗ U * B (g) on H A ⊗ H A and H B ⊗ H B have a multiplicity-free decomposition into irreps.More precisely, the tensor representation U A (g) ⊗ U * A (g) can be decomposed into l-irreps with l varying between 0 and 2j A .In other words, where H l is a (2l + 1)-dimensional Hilbert space carrying irrep l.Analogous statement holds for the output system B.This means that the symmetry-adapted basis of ITOs for the input and output systems have no multiplicities and are given by {T l m } m,l with m ∈ {−l, . . ., l}, l ∈ {0, . . ., 2j A }, (58a) {S l m } m,l with m ∈ {−l, . . ., l}, l ∈ {0, . . ., 2j B }. (58b) We note that we can choose T 0 0 and S 0 0 , corresponding to the trivial irrep l = 0, to be given by I A / √ d A and I B / √ d B , and T 1 m to be related to the angular momentum operators in the following way where and with analogous expressions for the output system B with S 1 m .Moreover, as the multiplicity spaces are 1-dimensional, the operators L λ (E) from Eq. ( 26) of Theorem 2 become scalars f λ (E).Therefore, the block diagonal decomposition of the Liouville representation of an SU(2)-covariant channel between irreducible systems has a simple block structure given by Employing the symmetry adapted basis of ITOs through Eq. ( 37), we can equivalently express the above by In other words, the covariant channel E transforms irreducible systems by simply scaling ITOs with irrepdependent magnitudes encoded in the scaling vector f (E).As a result, the initial state ρ A , given by Eq. (38) (without the sum over multiplicities α), is transformed into where we have used the fact that ITOs T 0 0 and S 0 0 are given by identities and, due to TP condition, f 0 (E) = 1.
At this point we know that the action of an SU(2)covariant channel E between irreducible systems is fully described by a scaling vector f (E) through Eq. ( 63), but to understand the relation between deviations from conservation laws and unitarity of E, we need to find the constraints on f (E).In particular, we will be interested in possible values of f 1 (E), since this number quantifies how much the angular momentum of the system changes under the action of E. To achieve this, we will look at the Jamio lkowski state J (E), enforce its positivity (to ensure CP condition), and tr B (J (E)) = I A /d A (to ensure TP condition), thus finding constraints on f (E) which ensure that it corresponds to a valid quantum channel.
Using Theorem 3, we find that the Jamio lkowski state is also block-diagonal, and the structure of the blocks is again very simple.This is because the tensor representation U B (g) ⊗ U * A (g) can be decomposed into L-irreps with L ∈ {|j A − j B |, . . ., j A + j B } and no multiplicities.In other words, where H L is a (2L+1)-dimensional Hilbert space carrying irrep L. As the multiplicity spaces are 1-dimensional, the operators J λ (E) from Eq. ( 29) become scalars, and thus we have Crucially, each J (E L ) corresponds to a valid Jamio lkowski state: it is clearly positive semi-definite, and the trace-preserving condition can be shown as follows.First, observe that for all g we have Then, since tr B J (E L ) ∈ B(H A ) commutes with an irrep U A (g) for all g, we can use Schur's Lemma 1 to conclude that tr B J (E L ) must be proportional to identity.Finally, normalisation of J (E L ) ensures that tr B J (E L ) = I A /d A .Moreover, since the supports of J (E L ) are disjoint, E L correspond to extremal channels, Clearly, so that in order to find constraints on f (E), we only need to find the values of f (E L ) for all L.More precisely, the set of allowed f (E) is then given by a convex set with extremal points given by f (E L ).
We will find f (E L ) by deriving the explicit action of E L on the basis elements {|j A , m j A , n|} with m, n ∈ {−j A , . . ., j A }. First, note that I L appearing in the expression for J (E L ) is given by Next, using Clebsch-Gordan expansion for the above total angular momentum states |L, k in terms of the angular momentum states of H A and H B , we write Now, employing the identity that holds for all X ∈ B(H A ), as well as the following two properties of Clebsch-Gordan coefficients, we arrive at Note that the action of an extremal channel E L can be physically interpreted as first splitting the original system with total angular momentum j A into two subsystems with total angular momenta j B and L (using Clebsch-Gordan coefficients), and then discarding the second subsystem.These extremal channels have been examined in detail in previous literature under the name of EPOSIC channels [13].
Finally, using Eq.(62) and noting that there exists m , n and k such that j B , n |S l k |j B , m = 0, we can write We emphasise that that the quantity above is independent of m , n and k.Now, by expanding T l k in the basis m, n ∈ {−j A , . . ., j A }, using Eq. ( 73), and employing Wigner-Eckart theorem, we can derive the following expression for f l (E L ): where j A ||T l ||j A and j B ||S l ||j B are reduced matrix elements independent of n , m or k.It simplifies significantly when j A = j B = j: We provide the step-by-step derivation of the above expressions in Appendix C, where we also show how to obtain the explicit formula for f 1 (E L ), which will be crucial for our analysis of spin-inversion and spin-amplification.Let us conclude this section by re-iterating the main result in the form of the following theorem.Theorem 7.An SU(2)-covariant channel E between two irreducible systems, carrying irreps j A and j B , is fully specified by a probability distribution p(E) of size 2 min(j A , j B ).Its action on X ∈ B(H A ) is then given by where x l k = tr T l † k X and f l (E L ) are specified by Eq. (75).

B. General decomposition of G-covariant channels
Let U A and U B be unitary representations of a compact group G acting on H A and H B , respectively.We are interested in quantum channels E : B(H A ) → B(H B ) that are symmetric under these actions.As explained in Sec.III B, the corresponding Jamio lkowski state J (E) will commute with the tensor product representation U B ⊗ U * A , which decomposes the Hilbert space Here, Λ is a subset of all non-equivalent irreducible representations labelled generically by λ that appear with multiplicities m λ (denoting the dimension of the multiplicity space).
From Theorem 3 we know that under such a decomposition the Jamio lkowski state of a symmetric channel has a block-diagonal structure: Note that in the above J λ (E) ∈ GL(C m λ ), and I λ acts as identity on the λ-irrep representation space H λ .Let us now define Since E is completely positive, we have J (E) ≥ 0 and thus ρ λ (E) ≥ 0.Moreover, the trace-preserving property of E implies that λ∈Λ p λ (E) = 1.Therefore, p λ (E) is a probability distribution and ρ λ (E) is a valid quantum state on GL(C m λ ).One should keep in mind, however, that there will be additional constraints on ρ λ (E) coming from the trace-preserving condition.We can thus write Now, recall that any state ρ λ (E) ∈ GL(C m λ ) can be viewed as a probability distribution over all pure state such that: where |ψ λ ∈ C m λ , integration is over all such pure states (according to the Haar measure) with dψ λ r E (ψ λ ) = 1 and r E (ψ λ ) ≥ 0. We can then define the following operators, which should be viewed as elements of B(H B ⊗ H A ) that are positive and have trace one.Therefore, any symmetric Jamio lkowski state J (E) can be written as follows This directly leads to the following decomposition of any G-covariant channel: Here, E λ ψ λ are CP maps corresponding to Jamio lkowski states J λ ψ λ .Note, however, that although the above resembles a convex decomposition over extremal channels E λ ψ λ , these are not necessarily trace-preserving.Therefore, the set of extremal G-covariant quantum channels may be much more complicated, e.g., with ψ λ ψ λ in Eq. ( 84) replaced by a mixed state.
More can be said about the structure of extremal channels under additional assumptions.The particular case we consider here is given by these symmetries for which representations U A and U B of a compact group G (acting on the input and output Hilbert spaces H A and H B ) are such that H B ⊗ H A , with the tensor product representation U B ⊗ U * A , has a multiplicity-free decomposition, Moreover, we will also require that U A is an irrep.One example of a group satisfying these assumptions is the SU(2) symmetry with the input system being irreducible, which we studied in detail in Sec.V A. For completeness, we remark that previous works [27] have fully characterised under what conditions tensor products of irreducible representations have a multiplicity-free decomposition for all connected semisimple complex Lie groups.
In particular, if G is a simple Lie group (e.g., SL(d)), then either U B or U * A must correspond to an irrep with the highest weight being a multiple of the fundamental representation.For example, for the group SU (3) with the fundamental irrep labelled by 3, we have a multiplicityfree decomposition 3 ⊗ 3 = 8 ⊕ 1.This stands to show that the assumptions can still include a large class of symmetries beyond the canonical SU(2) example, e.g., Ref. [15] studies covariant channels with respect to finite groups with multiplicity-free decomposition.
For groups satisfying these conditions, Eq. ( 82) simplifies significantly and takes the following form Here, by the same argument as in Sec.V A, p λ (E) is a probability distribution and each J (E λ ) is a positive operator satisfying tr B J (E λ ) = I A /d A .Therefore, each J (E λ ) will uniquely correspond to a CPTP map E λ : B(H A ) → B(H B ) and, since J (E λ ) act on orthogonal subspaces, they will be linearly independent operators.Equivalently, this ensures that E λ are extremal points of the set of G-covariant channels.We can thus characterise E λ in terms of Jamio lkowski states, Kraus operators and Stinespring dilation through the following theorem.
Theorem 8. Let G be a compact group with representations U A and U B acting on Hilbert spaces H A and H B .Suppose that U B ⊗ U * A is a multiplicity-free tensor product representation with non-equivalent irreps labelled by elements of a set Λ, and that U A is an irrep.Then, the convex set of G-covariant quantum channels E : B(H A ) → B(H B ) has |Λ| distinct isolated extremal points given by channels E λ for λ ∈ Λ.Each E λ can be characterised by the following: 1.A unique Jamio lkowski state 2. Kraus decomposition {E λ k } d λ k=1 such that: with E λ forming a λ-irreducible tensor operator transforming as k , where v λ k k are matrix coefficients of the λ-irrep.

A symmetric isometry W
Also, the minimal Stinespring dilation dimension for E λ is given by d λ .
The details on how to obtain characterisations 2. and 3. from 1. can be found in Appendix A.

C. Decomposition of U(1)-covariant channels
We now proceed to the simplest example of a compact group that does not satisfy the multiplicity-free condition -the U(1) group.As we will see in a moment, channels symmetric with respect to U(1) group do not satisfy this condition in the strongest possible way: they act trivially on the irrep spaces (since those are one-dimensional) and are fully defined by their action within the multiplicity spaces.In that sense, the example investigated in this section is the exact opposite of SU(2)-irreduciblycovariant channels studied in Sec.V A, where the action within multiplicity spaces was trivial and channels were defined by their action within irrep spaces.
The U(1) group has a single generator where g ∈ [0, 2π].For a finite-dimensional system2 described by a Hilbert space H A , the U(1) group can be related to time-translations by choosing the generator to be given by the system Hamiltonian H A , with E n A denoting different energy levels, and where we restricted ourselves to non-degenerate Hamiltonians for the clarity of discussion.Indeed, substituting J 1 A → H A and g → −t, we see that the group action, As U( 1) is an Abelian group, its irreducible representations are 1-dimensional, meaning that the symmetry adapted basis composed of ITOs satisfies It follows that we can choose and α enumerating multiplicities arising from the degeneracy of the Bohr spectrum of H A and H B , i.e., various pairs n, n satisfying the same λ = E n A/B − E n A/B .We consider a U(1)-covariant channel E, with the representations of the U(1) group on the input and output spaces, H A and H B , being given by U A (t) and U B (t), i.e., with the Hamiltonians of the input and output systems being H A and H B .Employing Theorem 2 we then get that the Liouville representation of E is block-diagonal, and from Eq. ( 37) we find that B and α, β enumerating degeneracies, i.e., various pairs of n, n and m, m with the same energy difference λ.
We see that the block λ = 0 describes the evolution of populations (in the energy eigenbases), while the remaining blocks describe the evolution of coherence terms between energy levels differing by λ.Therefore, L λ=0 (E) contains full information needed to study deviations from energy conservation induced by E, while L λ =0 (E) define how coherent E is, i.e., how close it is to a closed unitary dynamics.We note that the relation between L λ=0 (E) and L λ =0 (E) has played a crucial role in the previous studies on optimal processing of coherence under thermodynamic [28] and Markovian [29] constraints.Here, we will use this relation to constrain the unitarity of a general U(1)-covariant channel inducing energy flows (deviating from energy conservation) described by a given stochastic matrix.Since L λ=0 (E) is crucial for our studies, we will use a shorthand notation P E for it, and note that it is a d B × d A stochastic matrix, P E mn ≥ 0 and m P E mn = 1.As our aim is to study the relation between deviations from conservation laws and unitarity of U(1)-covariant channels, we need to understand what are the constraints on L λ (E).To answer this question we will look at the Jamio lkowski state J (E) and, by enforcing its positivity and tr B (J (E)) = I A /d A , we will find constraints on matrices L λ ensuring that they correspond to a valid CPTP map.From Theorem 3 we get that the Jamio lkowski state is also block-diagonal, where The positivity of J (E) is now equivalent to the positivity of J λ (E) for all λ, while the partial trace condition is fulfilled automatically as long as P E is a stochastic matrix.Importantly, the diagonal of J λ (E) is given by , while the off-diagonal terms describe transformation of coherences.One can now construct extremal U(1)covariant channels by simply coherifying any stochastic matrix Γ to a quantum channel with the constraint of preserving the block-diagonal structure [30].More precisely, for every stochastic matrix Γ and a set of phases {φ λ,m }, one can construct an extremal U(1)-covariant channel E Γ,φ with the Jamio lkowski state given by with ) where Γ describes P E Γ,φ and the same notation applies to its elements Γ mn .It is a straightforward calculation to show that the corresponding map is CPTP.Moreover, since its Jamio lkowski state is proportional to a projector on each block, it is extremal.
We want to note, however, that the above construction in general does not produce all extremal U(1)-covariant channels.As a counterexample, consider the following Jamio lkowski state Since the above is extremal on each block λ = λ * , the possibility of decomposing it as a convex combination of Jamio lkowski states from Eq. ( 102) is equivalent to the possibility of decomposing ρ λ * as In other words, it would need to hold that every density matrix of size d can be decomposed into a convex combination of pure states with the same diagonal.This, however, is not true in general (it holds for d = 2, but counterexamples can be found already for d = 4).

A. Setting
The scenario investigated in this section is as follows.We consider input and output systems, described by Hilbert spaces H A and H B , to be spin-j A and spin-j B systems.We denote the spin angular momenta operators (with respect to a Cartesian coordinate frame) by and analogously for J B .These are traceless and for every k ∈ {x, y, z} satisfy tr(( where J A was defined in Eq. ( 60).Analogous conditions hold for system B. We recall that these spin operators are generators for the SU( 2 59).Now, for the input and output state, ρ ∈ B(H A ) and E(ρ) ∈ B(H B ), we can define spin polarisation vectors, P(ρ) and P(E(ρ)), to be given by expectation values of the spin operator along different Cartesian axes: and similarly for the system B with ρ replaced by E(ρ).
Our aim is to investigate operations that isotropically invert or amplify the spin operator, so that under their action the polarisation vector scales with either some negative factor κ − , or a positive factor κ + > 1.In particular, we want to determine channels S − and S + , representing the optimal spin-inversion and spin-amplification, which are those that achieve the largest values of |κ − | and κ + : Equivalently, S ± may be defined in terms of their action on the generators: First, we will take the above equations as really defining S ± , without specifying their action outside of the subspace spanned by the generators J k A .This will, in principle, correspond to a large class of operations that we need to optimise over.However, since S ± acts isotropically on all states, in the next section we will show that without loss of generality one may restrict considerations to SU(2)-covariant channels.This will allow us to employ results of Sec.V A to determine optimal inversion and amplification factors κ ± , and to relate κ − to the maximal allowed deviation from conservation law under covariant dynamics.Finally, we will compare focus on the decoherence induced by the optimal inversion channel by comparing the action of this channel with the action induced by time-reversal symmetry.

B. Optimal transformations of spin polarisation
We want to analyse channels E : B(H A ) → B(H B ) that send P (ρ) to κP (ρ) for all ρ and some independent real constant κ, while performing arbitrary transformation on the other irreducible subspaces (ITOs).As we will now show, for every such E there exists an SU(2)covariant channel that has the exact same action on the polarisation vector.By assumption, Now, with U A denoting the SU(2) representations on B(H A ), recall that the angular momentum operators transform under rotations as where v 1 k k (g) are matrix entries of the 1-irrep.Analogous statement holds for system B. Therefore, it follows that Using the cyclic property of the trace and the fact that the above must hold for all ρ, so in particular for U g A (ρ), we arrive at or equivalently: We note that the above could also be simply deduced from the fact that P transforms under SU(2) as a threedimensional vector in real space, so that for all g ∈ SU(2) and all ρ we have Next, by taking the group average and noting that P is linear (since it is defined through trace in Eq. ( 108)), we obtain where G[E] is the twirling of E over all rotations, The twirled channel G[E] is SU(2)-covariant (by construction), and it has the same scaling factor κ as E. Therefore, one may assume without loss of generality that the optimal spin-inversion and amplification operations are symmetric under SU(2).In Sec.V A we have fully characterised SU(2)-covariant quantum channels for irreducible systems and we will now employ these results.First, recall that a symmetric channel E : B(H A ) → B(H B ) acts on any ITO {T l k } l,k by a scaling factor depending only on the particular irrep (and the channel itself) such that Taking into account the particular normalisation of the spin operators it follows that: Moreover, recall that every such SU(2)-covariant E decomposes into extremal channels according to Eq. ( 67), which results in where f 1 (E L ) are the scaling factors explicitly given by Eq. ( 77).Now, we can compare the transformation of angular momentum operators under a general covariant channel, Eqs.(120)-(121), with the transformation under spininversion and spin-amplification, Eq. (110).We see that every SU(2)-covariant channel can act as a spin-inversion or amplification with as long as κ + > 1 and κ − < 0. Our aim is thus to maximise and minimise the above expression over all probability distributions p L .Since we are optimising over a convex region, the optima will be attained by one of the extremal points so that for some L ± .These can be easily found, as we derived explicit expressions f 1 (E L ).
We thus arrive at: Theorem 9.The maximal spin polarisation inversion, P (ρ) → κ − P (ρ) with κ − < 0, is achieved by an SU(2)irreducibly extremal channel E (j A +j B ) .The inversion factor κ − is given by It follows that the maximal spin-inversion is achieved by the extremal channel that requires the largest environment to be realised.Indeed, for every extremal channel E L , its minimal Stinespring dilation (and thus the minimal number of Kraus operators) has dimension 2L + 1.Consequently, this means that the larger the environment, the more we can invert the spin.Note that in the classical macroscopic limit of an input and output system given by a massive spin, j A = j B → ∞, we get κ − → −1, corresponding to perfect spin-inversion.While for finite-dimensional systems quantum theory does not allow for perfect spin-inversion, P → −P, the above result yields fundamental limit on maximal spin-inversion.
Moreover, the optimal spin-inversion coincides with the channel leading to the largest allowed deviation from the conservation law under the constraint of SU(2) symmetry.To see this note that the total deviation resulting from the action of E on a given input state ρ (defined in Eq. ( 51)), can be expressed by Using the fact that covariant dynamics can only scale ITOs, we get and thus the deviation is maximised for smallest negative κ, which is specified by Eq. ( 124).From the equation above it is clear that also average total deviation, ∆(E), will be maximised by the optimal spin-inversion channel.Of course, since we deal with symmetric channels, this deviation can come only for the price of decoherence (as the conserved charge can only come from an incoherent environment).In the next section, we will quantify this decoherence by comparing the action of the optimal spin-inversion channel with the transformation induced by time-reversal symmetry; while in Sec.VII we will analyse in detail the trade-off between deviations from conservation laws and decoherence for general SU(2)-covariant operations.Finally, we can obtain an analogous bounding result for spin amplification, captured by the following theorem.
Theorem 10.The maximal spin polarisation amplification, P (ρ) → κ + P (ρ) with κ + > 1, is achieved by an SU(2)-irreducibly symmetric extremal channel E (|j A −j B |) .The amplification factor κ + is given by: We remark that upper bounds on κ + have been previously reported in Ref. [23], where the authors used resource monotones based on modes of asymmetry to show that κ + ≤ f (j B )/f (j A ) with f (j) := j + 1/2 : integer j, j(j + 1)/(j + 1/2) : half integer j. (128) Note that, according to Theorem 10 that provides the optimal amplification channel explicitly, these bounds are loose, i.e., the upper bound cannot be achieved by any SU(2)-covariant channel.In this sense our result can be seen as an ultimate improvement over the previously known bounds.

C. Optimal spin-inversion and time-reversal symmetry
So far we have considered the action of a channel on spin polarisation vector as the defining property of the spin-inversion channel.We have thus focused on the maximal deviation from the conservation law, but ignored the decoherence induced by such a channel, which is described by the action of the channel on the remaining ITOs.Here, we will quantify this decoherence by comparing the action of the optimal spin-inversion channel to the action of a passive symmetry that naturally realises spin-inversion -the time-reversal symmetry T .
Under the action of T : B(H A ) → B(H A ) the spin of a single particle flips sign and, generally, an odd number of particles will experience a sign change, while an even number will not.This manifests itself at the level of ITOs, which are mapped according to whether they correspond to even or odd dimensional irreducible representations of the rotation group: This fully captures the action of time-reversal on general mixed states of spin-j A systems described by the Hilbert space H A .In particular, the spin degrees of freedom under time-reversal will acquire a minus sign: Therefore, for a single particle time-reversal symmetry induces perfect spin reversal, as for any ρ ∈ B(H A ) the spin polarisation vector satisfies P (T (ρ)) = −P (ρ).Moreover, T does not induce any decoherence, since it leaves the eigenvalues of ρ unchanged.It is thus meaningful to compare the optimal physical spin-inversion channel E (2j A ) from Theorem 9 with the perfect unphysical spin-inversion operation realised by time-reversal symmetry T .We will see that E (2j A ) , although it inverts spin polarisation almost perfectly in the limit of large j A , is always far away from realising T , and thus induces unavoidable decoherence as expected.
In order to measure the distance between E (2j A ) and T let us introduce the concept of a spin-coherent state.It is simply given by a rotation of |j A , j A , the state with maximal angular momentum along the z axis.Suppose that the group element g ∈ SU(2) is characterised by the Euler angles θ, φ, corresponding to a spatial direction n.Then the spin-coherent state associated to this direction is given by: The behaviour of spin coherent states under time-reversal symmetry is particularly simple and reads In order to quantify how much the optimal spininversion channel E (2j A ) resembles the passive symmetry transformation T we will employ the notion of quantum fidelity, Namely, we will calculate the fidelity F between the outputs of the two channels averaged over all input spincoherent states.Notice that the fidelity between two states is a unitarily invariant measure, so that, and, since both E (2j A ) and T are SU(2)-covariant, it follows that the considered fidelity remains the same for all spin coherent input states.Therefore, it suffices to analyse the fidelity for the input state |j A , j A , i.e., Now, we can use the explicit form of E (2j A ) given in Eq. ( 73) to arrive at Finally, employing Clebsch-Gordan coefficients identities we obtain The above fidelity is monotonically decreasing as a function of j A and in the limit j A → ∞ it converges to 1/2.Therefore, despite the fact that for macroscopic spins it is possible to almost perfectly invert their polarisation vector, the channel that achieves this is far from realising time-reversal symmetry.We remark that the above calculation only assumes that the action of T on the spin-coherent state |j A , j A gives |j A , −j A and that it is rotationally invariant.Therefore, the same result will hold for a general perfect and unphysical spin-inversion operation which satisfies these two constraints (without committing to the full exact form that the time-reversal operator takes).Moreover, note that the rotational invariance and linearity ensures that the expression for F remains unchanged for any state in the convex hull of spin-coherent states.

VII. TRADE-OFF RELATIONS BETWEEN CONSERVATION LAWS AND DECOHERENCE
Building up on the results developed so far, we now address the core questions of interest: how much can open symmetric dynamics deviate from conservation laws?Do small perturbations from closed symmetric dynamics result in small corrections to the conservation laws?When does the converse also hold?
Our aim is therefore to analyse when each of the following two qualitative statements holds given an a priori symmetry principle: • If E is close to a symmetric unitary then the average total deviation from conservation law, ∆(E), is small.
• If the average total deviation ∆(E) is small then E is close to a symmetric unitary.
Whenever both of the above properties hold for any dynamics with the appropriate symmetry, we say that the conservation laws are robust with respect to decoherence.Quantitatively, we can analyse such robustness by deriving bounds on the average deviation induced by a channel in terms of its distance from a symmetric unitary process.
In what follows, we first derive general upper bounds on the deviation in terms of the diamond distance (for arbitrary dimension of input and output spaces, d A and d B ) and unitarity (for d B ≤ d A ), showing that the first property holds in general.Then, we argue why a lower bound does not need to exist for a general group G, and so the second property does not need to hold.Nevertheless, we show that for symmetries with multiplicity-free decomposition, the lower bound can also be derived for d A = d B , and thus conservation laws are robust under decoherence in such cases.Finally, we analyse in detail the two special examples investigated in Sec.V: SU(2)irreducibly-covariant channels and U(1)-covariant channels.

A. Upper bounds on deviating charges for G-covariant open dynamics
Before we present our main result upper bounding the average total deviation ∆(E) as a function of departure from unitarity (1 − u(E)), we want to present a simple argument showing that open dynamics that is close to symmetric unitary (isometry) must approximately conserve relevant charges.Consider ρ ∈ B(H A ) and a G-covariant channel E : B(H A ) → B(H B ) with the symmetry generated by {J k A } n k=1 for the input system and {J k B } n k=1 for the output system.Now, take any isometry W : H A → H B that is symmetric, i.e., W J k A = J k B W . Since the conservation laws hold under a dynamics generated by W , we have Using Hölder's inequality for the Hilbert-Schmidt inner product, with B 1 := tr( √ B † B) and A ∞ the operator norm, we obtain the following bound: where W(•) = W (•)W † .Thus, the total deviation for a given input state ρ A is bounded by Finally, we can get a state-independent bound by employing a diamond norm, so that we arrive at the bound for the average total deviation Operationally the above can be interpreted as follows: the more indistinguishable a given covariant channel becomes from any symmetric isometry, the smaller the deviations from conservation laws.obviously, the above simple analysis has significant drawbacks.Not only is the diamond norm particularly difficult to calculate, but also Eq. (142) involves either an unknown symmetric isometry W, or a minimisation of the quantity E − W over all such isometries W. The latter will generally be difficult to estimate from the properties of the channel E alone, leading to very loose upper bounds on the average total deviation.For these reasons, in the following theorem we provide an explicit inequality that captures robustness of conservation laws in terms of the unitarity of a symmetric channel.
Moreover, the above also holds for d A < d B whenever tr(E(I/d A ) 2 ) ≥ 1 d A .To prove the above theorem one starts from Eq. ( 82) that yields the general decomposition of a G-covariant map into a convex mixture of CP maps E λ with probabilities p λ (E).Employing this decomposition and Lemma 6, one can then lower bound the deviation from closed dynamics, (1 − u(E)), with a dimensional constant times (1 − p λ=0 (E)) 2 .Next, one notes that E λ=0 conserves charges (generators), and thus using standard inequalities (e.g., the triangle inequality) the deviation can be upper bounded by a dimension-dependent constant times (1−p λ=0 (E)) 2 .Finally, one combines both inequalities to bound ∆(E) with (1 − u(E)) as in Eq. ( 143).The details of necessary calculations can be found in Appendix D.

B. Lower bounds on deviating charges for G-covariant open dynamics
We would now like to find a lower bound on the average total deviation in terms of unitarity.First, however, we need to note that decoherence does not need to lead to the deviation from conservation law.In other words, there may be open (non-unitary) symmetric dynamics that nevertheless conserves charges (generators) for all input states.To illustrate this, let us start with the following semi-trivial example of a non-unitary symmetric dynamics E for which all conservation laws relevant for the symmetry hold.Consider a two-qubit system where the first qubit transforms under the 1/2-irrep of SU(2) and the second transforms trivially.The conserved charges generating the symmetry are the spin operators on the first system.Let E AB (ρ ⊗ σ) := ρ ⊗ E B (σ).This is covariant under the symmetry, the conservation laws hold for all states, however it is not a unitary operation as we are free to choose any CPTP E B on system B.
More generally, there may exist whole families of nontrivial symmetric channels that are not unitary, but preserve conserved charges for all input states.For example, it is relatively simple to find such a family among unital covariant channels.Theorem (4.25) from Ref. [6] tells us that for a unital CPTP map E : B(H A ) → B(H A ) with Kraus operators {K i } i we have E(X) = X if and only if [X, K i ] = 0 for all i.Recall also that any symmetric channel E admits a Kraus decomposition consisting of ITOs {E λ,α m } λ,m,α , where λ labels irreducible representations in B(H A ) of multiplicity α and vector component m.Then, it follows that for all symmetry generators J k A .Now, since we assumed that E is unital CPTP map, also E † is a unital CPTP map.Thus, we can use the result quoted above and conclude that E † (J k A ) = J k A if and only if [E λ,α m , J k A ] = 0 for all λ, m, α and k.However, E λ,α m transform as ITOs and only λ = 0, corresponding to the trivial representation, commutes with the generators.Therefore, for unital symmetric channels E, conservation laws hold if and only if E takes the general form: where each Kraus operator E 0,α commutes with the group action.In general it may also be possible for conservation laws to hold for non-unital operations, but a full characterisation of the dynamics for which this happens remains open.
As the examples above conserve charges despite decoherence by acting on the multiplicity spaces of the trivial representation λ = 0, one could hope that for groups with multiplicity-free decomposition such a situation will be impossible (and so the conservation law would be robust to decoherence).However, this is not the case.To see this, recall that in Sec.VI we found the extremal SU(2)irreducibly covariant channel E |j A −j B | that allowed for spin amplification whenever d B > d A .At the same time, we showed that there also exists an optimal spin reversal channel E j A +j B .Thus, one can always find a parameter q ∈ [0, 1] such that qE |j A −j B | + (1 − q)E j A +j B preserves all spin components, while at the same time being far from unitary evolution.
The above discussion illustrate that probing conservation laws for a physical realisation of a symmetric dynamics is usually not sufficient to decide if there are decoherence effects present.In other words, robustness of conservation laws does not occur for all types of symmetries.Nevertheless, there are particular conditions that guarantee a certain robustness to conservation laws.In such cases, approximate conservation laws hold if and only if the dynamics is close to a unitary symmetric evolution.In particular, for channels with equal input and output dimensions d A = d B , whenever B(H A ) contains a single trivial subspace then there is no symmetric channel other than identity for which conservation laws hold.This is the case for example when H A carries an irreducible representation of SU (2).More generally, however, we have the following theorem that provides lower bounds on the deviation from conservation laws in terms of the unitarity.
Theorem 12. Let G be a connected compact Lie group with unitary representation U A acting on a Hilbert space H A , and generated by traceless generators {J k A } n k=1 .Moreover, assume that B(H A ) has a multiplicity-free decomposition in terms of irreducible representations.Then, for every G-covariant quantum channel E : B(H A ) → B(H A ) the following holds: where K is a constant independent of E, defined by with f (λ) being constant coefficients such that the extremal isolated channel The proof of the above theorem can be found in Appendix E.

C. Bounds on deviating charges for SU(2)-covariant open dynamics
We now turn to investigating robustness of conservation laws for SU(2)-irreducibly-covariant channels.We focus on a particular case of covariant quantum channels between spin-j systems, i.e., for j A = j B = j.In this case it is possible to deduce both upper and lower bounds on average total deviation in terms of unitarity.One of the reasons for this is that dissipation, as given by a symmetric channel E that is not a unitary, cannot hide in the multiplicity subspace of the trivial representation, as U A ⊗ U * A has a multiplicity-free decompositions into irreps.

Expressions for unitarity and deviations
The structure of general SU(2)-irreducibly-covariant channels E : B(H A ) → B(H B ) presented in Sec.V A gives a simple way to calculate their unitarity.Employing Lemma 6, using the decomposition of the Jamio lkowski state given in Eq. ( 65) and the fact that irreducibly-covariant channels are unital (so that where p(E) characterises a given SU(2)-covariant channel according to Eq. (67).In the particular case when j A = j B = j, so that both input and output dimensions d = 2j + 1, the above yields Now in order to get an expression for the average deviation from a conservation we first look at where k ∈ {x, y, z} correspond to the spin angular momentum operators J x , J y , J z .We start by noting that, due to Eq. ( 62) and the fact that ITOs are orthonormal, we have Next, using the relation between angular momentum operators and ITOs from Eq. (59), we can re-write the above expression to arrive at where we recall that J B = j B (j B + 1)(2j B + 1) and analogously for input system A. Next, we use convex decomposition of E into extremal channels E L , Eq. ( 67), to get However, we have determined specific closed formulas for f 1 (E L ) in Eq. ( 77), so that combining with the above relations we end up with with Finally, the spin angular momenta satisfy the following relations (and similar ones for system B): Combining the above with Eq. ( 152) and substituting to general expression for the average total deviation, Eq. ( 55), leads to In the particular case when j A = j B = j the above yields (156)

Deriving trade-off relations
We will now show how the unitarity and deviations from conservation laws are related, and obtain both lower and upper bounds on the average deviation from a conservation law of spin angular momenta under a rotationally invariant irreducible channel in terms of its unitarity.
Theorem 13.Let E : B(H A ) → B(H A ) be an SU(2)irreducibly covariant quantum channel acting on a j-spin system.Then, the average total deviation ∆(E) from conservation law for spin angular momenta is bounded by the unitarity of the channel u(E) via the following trade-off inequalities: Proof.First, using Eq. ( 156), we note that given a fixed p 0 the deviation ∆(E) is maximised for p 2j = 1 − p 0 .Therefore, resulting in the following bound To shorten the notation we will now use d = 2j + 1 and j simultaneously.Using Eq. ( 147) we have the following series of equalities and inequalities: where the second inequality comes from the fact that the sum of squares of positive numbers is upper bounded by the square of the sum, and the final one from Eq. (159).
On the other hand, using Eq. ( 156) again, we note that given a fixed p 0 the deviation ∆(E) is minimised for resulting in the following bound To shorten the notation we will use d = 2j + 1 and j simultaneously, and introduce A := 2j(4j + 1).We then have the following series of equalities and inequalities: with the second inequality coming from the fact that the sum of squared probabilities, given a constraint on the total probability, is minimised for uniform distribution; and the final inequality coming from Eq. (162).

Examples
Consider first the simplest example of a covariant channel for j = 1/2.In this case the unitarity is given by and the deviation by A straightforward calculation then yields a direct relation between u(E) and ∆(E), while the bounds from Theorem 13 read We present the above dependence and bounds in Fig. 4, where we also plot the general upper bound from Theorem 11.
The next simplest case concerns spin-j system with j = 1.We then have where p 2 (E) = 1 − p 0 (E) − p 1 (E) and the deviation is given by: Unitarity u(E) and deviation ∆E are no longer directly related, but they constrain each other, so that only some pairs [∆(E), u(E)] are realised by SU(2)-covariant channels.Our bounds then take the form and again we plot them in Fig. 4 together with the general upper bound and possible pairs [∆(E), u(E)].

D. Bounds on deviating charges for U(1)-covariant open dynamics
We now turn to our final example of U(1)-covariant dynamics and the corresponding trade-off between deviations from energy conservation and unitarity of the channel.Throughout this section we employ the notation introduced in Sec.V C while studying the convex structure of U(1)-covariant channels.For simplicity, we focus on the input and output systems of the same dimension, d A = d B = d, and described by the same Hamiltonian H A = H B = H.As we will shortly explain, in this case it is impossible to lower bound unitarity u(E) given the deviation ∆(E), and thus our aim is to upper bound the unitarity u(E) given the deviation ∆(E), i.e., to find the minimal allowed departure from a closed symmetric dynamics that can explain a given deviation from energy conservation.

Expressions for unitarity and deviations
Substituting the decompositions given in Eq. (100) to Eq. ( 44), one obtains the following expression for unitarity of a general U(1)-covariant channel E: where describes how far P E is from a bistochastic matrix, i.e., b E = 1 when P E is bistochastic and b E > 1 otherwise.The expression for the average total deviation ∆ is given by Eq. (55), Moreover, since H is diagonal in the energy eigenbasis, the expression for E † (H) only involves λ = 0 block, and so δH can be easily calculated explicitly.More precisely, one gets

Deriving trade-off relations
First of all, we note that the deviation ∆(E) depends only on P E , while the unitarity u(E) depends both on P E (forming diagonals of J (λ) (E)) and on L (λ =0) (E) (forming the off-diagonal terms of J (λ) (E)).Therefore, it is impossible to lower bound unitarity given the deviation.To see this more clearly, consider the following family of partial dephasing channels (which are U(1)-covariant): with Clearly, P Dp mn = δ mn and so ∆(D p ) = 0.However, the unitarity varies between 1 (for p = 0) and 1/(d + 1) (for p = 1).This is in accordance with our discussion in Sec.VII B concerning general non-existence of the lower bounds.Thus, we focus on deriving upper bounds.
We start by noting that each purity term in Eq. ( 171) is upper bounded by tr J (λ) (E) (this simply corresponds to J (λ) (E) being unnormalised projectors).Using this observation, as well as the fact that b E ≥ 1, we get with corresponding to the (unnormalised) probability of energy λ flowing into the system due to the action of E. Note that Eq. ( 177) yields a bound on unitarity that is expressed purely in terms of P E .Now, the crucial point is that for λ = 0 we have q λ ≤ g, with g denoting the largest number of pairs of energy levels separated by the same energy difference.The minimal value of g is 1, corresponding to a Hamiltonian H with non-degenerate Bohr spectrum, while the maximal value is (d − 1), achieved for a Hamiltonian with an equidistant spectrum.Since λ q λ = d, this means that the upper bound in Eq. (177) will be strictly smaller than 1 if q 0 < d.In other words, as soon as there is any energy flow induced by E (captured by P E nn < 1 for at least one n), unitarity u(E) will be strictly smaller than 1.
Let us now relate this observation to a concrete bound on u(E) involving ∆(E).First, we introduce the width of the energy spectrum: This allows us to get the following bound, with the second inequality coming from the fact that the sum of squares of positive numbers is upper bounded by the square of the sum.Combining these two bounds together we arrive at Next, we will rewrite Eq. ( 177) in a more convenient form as For a fixed q 0 the right hand side of the above equation is maximised when for some λ we have q λ = d − q 0 and q λ = 0 otherwise.However, this may not be possible due to a constraint q λ ≤ g.Thus, we need to consider two separate cases.First, assume that q 0 ≥ (d − g), so that d − q 0 ≤ g and the constraint is satisfied.We then have where the final inequality comes from Eq. ( 182).On the other hand, if q 0 < (d − g), then we can upper bound the unitarity by choosing the maximal allowed value q λ = g', and the remaining energy flows to q λ = d − g − q 03 .This means that As we do not know what the value of q 0 really is (we only know what ∆(E) is), we need to choose the weaker of the two bounds and thus we end up with (186)

Example
Consider a qubit system with unit energy splitting, Ẽ = 1.Average total deviation is then given by ∆ Trade-off between the deviation from energy conservation, ∆(E), and unitarity u(E) for U(1)covariant channels Each blue dot represents [∆(E), u(E)] pair for a qubit channel with fixed P E and optimal unitarity (the two parameters defining P E , 0 ≤ P E 00 , P E 11 ≤ 1, are taken as points from the lattice [0, 1] × [0, 1] with lattice constant 0.02).The orange solid line is the upper bound from Eq. (190).
while the optimal unitarity (obtained by choosing the blocks of the Jamio lkowski matrix to be unnormalised projectors) for a fixed matrix P E is given by u In Fig. 5 we present the region of all achievable pairs [∆(E), u(E)] (i.e., for each matrix P E we plot the corresponding deviation from energy conservation and the optimal unitarity of the quantum channel transforming energy eigenstates according to P E ), together with our bound from Eq. (186) that for this example reads

VIII. CONCLUSION AND OUTLOOK
We have studied the relationship between symmetry principles and conservation laws for irreversible dynamics that goes beyond Noether's principle.We established that the two questions posed in the introduction are fundamentally related.On the one hand, we provide the optimal active transformation approximating spin polarisation inversion, but this turns out to be the symmetric channel that achieves maximal deviation from the conservation law of spin angular momenta.Both of these limitations arise as fundamental constraints imposed by quantum theory on the connection between symmetry principles and conservation laws.At the core of these statements lies the convex structure of symmetric channels.Generally, classifying the structure of extremal (symmetric) channels [31] is a difficult problem that remains open in the general setting [32].For particular symmetries, the structure simplifies significantly and in several situations all extremal channels become isolated, forming a polyhedron.This was the case of symmetries described by irreducible representations of SU (2) analysed in detail in Ref. [13,14], but it can occur also for finite groups [15] and Weyl groups [33].Channels that are symmetric under an irreducible representation of some compact group are of particular importance in quantum information as their classical capacity is related to their minimal output entropy [11,15,34,35].This simpler structure was also crucial to our analysis of the robustness of conservation laws under symmetric irreversible dynamics.
We have restricted our analysis of Noether's principle to symmetric dynamics described by completely positive maps.Violations from conservation laws can occur in a variety of situations, including classical systems with dissipation leading to modified conserved currents, extensions of Noether's theorem for classical Markov processes [2,3].In using the formalism of CPTP maps, there is an assumption that the quantum system of interest is initially fully decoupled from its environment.A further direction to explore can be the situation when the system is coupled to the environment.This would lead to a local dynamical map corresponding to non-CP noise.We expect the stability of conservation laws under such dynamics to be difficult to characterise solely in terms of the local dynamics on the main system; we conjecture that in such case the upper bounds on deviation from conservation law in terms of unitarity of the (now non-CP) dynamics will no longer hold, due to a strong dependence on the initial system-environment interaction.
Finally, we speculate on the relevance of this work to relativistic quantum information theory, where decoherence induced by relativistic effects [36] can have an impact on probing conservation laws for the quantum systems involved.
comes from the fact that the Haar measure is an unitarily invariant measure.In this manner we can write u(E) in d A different ways.For instance it also holds that: The above holds for all basis states so we have in total d A equation.Summing all together we notice that we obtain the following In the above each term tr(E(ψ j )E(ψ k ) appears d A − 2 times with coefficient 1) and twice with coefficient 2 d A the latter arising from the equations for which i = j and i = k and the former from the rest of the equations, where we consider the j, k label to include ψ as well.Putting it all together − 4 The quadratic terms tr(E(ψ i ) 2 ) will appear once with coefficient d A −1 d A and then with coefficient 1) in each of the rest d A − 1 equations, which sums up to one.Now it is always true that tr(E(ψ i ) 2 ) ≤ 1 for all i and also since E is a CPTP map then E(ψ i ) is a positive operator so tr(E(ψ j )E(ψ k )) ≥ 0. Therefore it follows that: with equality holding for tr(E(ψ) 2 ) = 1, that is whenever E(ψ) is a pure state or equivalently when E is an isometry.⇐= Conversely, if E(X) = V XV † for some isometry V : H A → H B then V † V = I A and in this case we get where we have used the fact that ψd ψ = I/d A .Collecting terms it follows that if E is an isometry then u . However trace preserving condition implies that tr(E(I/d Lemma 16.Given a channel E : B(H A ) → B(H B ) then the unitarity can be equivalently expressed by: Proof.(i) Directly from the definition of unitarity we get: We note that when the Haar measure over pure states is properly normalised then the following hold ψd ψ We also have the following relation tr(ρ 2 ) = tr(SW AP A ρ ⊗ ρ) for all ρ ∈ B(H A ).One can similarly define the SWAP operator for system B. Therefore it follows that the average output purity is the purity of the Jamiolkowski operator: where Putting everything together it follows that tr(E(ψ) Similarly we have by linearity that: Therefore we get that the unitarity is given by and re-arranging we obtain (ii) To show the second part we just need to check that tr(J [E] 2 ) = tr( Ẽ(I/d A ) 2 ).First suppose that V : n=1 is an orthonormal basis for system B and that SW AP B = n,m |e n |e m e m | e n | The result then follows from the following argument:  In Sec.V A we have seen that the set of SU(2)-irreducibly-covariant channels between spin-j A and spin-j B systems is fully characterised by its extremal points E L : B(H A ) → B(H B ) with L ranging from |j A − j B | to j A + j B in increments of one.Since the input and output spaces carry irreducible representations j A and j B of SU (2), this means that the decomposition of the operator spaces into irreducible components is multiplicity-free, and therefore the results on the structure of the corresponding Liouville operators holds.For each extremal channel E L there is a unique vector f (E L ) of coefficients that fully determines it.Moreover, for SU(2) symmetries we can always construct basis of irreducible tensor operators that are Hermitian, which implies that these coefficients are real for any covariant quantum channel.Therefore, each of the vectors f (E L ) represents one of the extremal points that form a simplex in R d , where d = 2 min(j A , j B ).
Since we have a full characterisation of the channels E L , we can give closed form formulas for the vectors f (E L ) in terms of Clebsch-Gordan coefficients.In doing so, we will make use of the Wigner-Eckart theorem.As before, let {T λ k } k,λ and {S µ k } k,λ be ITO bases for B(H A ) and B(H B ), respectively.We have that E L (T λ k ) = f µ (E L )S µ k δ µ,λ for any L, λ, µ and k.The vector f (E L ) has entries f λ (E L ) with λ ranging from 1 to min(2j A , 2j B ); for λ = 0 trace-preserving condition implies that f 0 (E L ) = 1 2j B +1 is constant for all covariant channels, so we will not include it further into the vector definition of f (E L ).
Concerning the angular momentum states that form the basis for H A and H B as in Sec.V A, for any λ-irrep there exists labels m , n and k such that j B , n |S λ k |j B , m = 0. Therefore we can conveniently re-write each coefficient as: where we re-iterate that at the core of our analysis is that the quantity above is independent of m , n and k, and this is solely as a consequence of covariance of E L .The numerator can be written in an equivalent form by a basis expansion Therefore, by using the specific action of E L on angular momentum states given in Eq. ( 73) we obtain that: In particular, since the above factor has no dependence on the labels m , n and k, without loss of generality we can take k = 0, m = n = j B .We thus end up with the following expression: j A , j B + s; λ, 0|j A , j B + s j B , j B ; λ, 0|j B , j B j B , j B ; L, s|j A , j B + s 2 .(C6) In the particular case when the input and output spaces have the same dimension and both carry the same irrep of SU(2), j A = j B = j, we obtain the following: f λ (E L ) = L s=−L j, j + s; λ, 0|j, j + s j, j; λ, 0|j, j j, j; L, s|j, j + s 2 .(C7) 2. Maximal inversion and amplification of spin polarisation vector for SU(2)-covariant channels Here, we will characterise the range of values that the coefficient f 1 (E L ) takes while varying over all extremal channels L. This factor corresponds to how much the spin polarisation can scale (up or down) under a covariant operation.As we will see, due to the particular choice of irrep λ = 1, we can significantly simplify the expressions with Clebsch-Gordan coefficients appearing in Eqs.(C6)-(C7).We will first analyse the simpler case of same input and output dimension, and then proceed to the general case.In the former case, we find that while the spin polarisation cannot increase, the spin can be inverted up to a factor that is always greater than −1.In other words, we show that − j j+1 ≤ f 1 (E L ) ≤ 1, where the upper bound is attained for L = 0, i.e. the identity channel; and the lower bound is attained for L = 2j, i.e. the extremal channel with the maximal number of Kraus operators.In the latter case, when the output dimension is larger than the input one, we will show that spin polarisation vector can actually be amplified.

a. Input and output systems of the same dimension
From the explicit formula for f λ (E L ), Eq. (C7), we have that: L s=−L j, j + s; 1, 0|j, j + s j, j; 1, 0|j, j ( j, j; L, s|j, j + s ) It turns out that the above expression can be easily evaluated in terms of products of binomial coefficients, so that We can compute each of the two sums above separately by using combinatorial identities, to obtain a closed form formula for f 1 (E L ): Therefore, under any SU(2)-covariant channel, the spin polarisation can either remain the same (whenever L = 0, which corresponds to the identity channel), decrease by 0 ≤ f 1 (E L ) ≤ 1, or get inverted by f 1 (E L ) ≤ 0. However, in this scenario the spin polarisation will never increase.The maximal deviation from a conservation law is achieved by the extremal channel L = 2j which also achieves the maximal spin inversion of polarisation: b. Input and output systems of different dimensions We now proceed to the case j A = j B .From Eq. (C6) for λ = 1 we have the following: For operators on the carrier space H A for the j A -irrep (and similarly for j B ), the decomposition of B(H A ) contains each irreducible representation with multiplicity at most one.For any j A > 0 the 1-irrep will appear once, and the corresponding subspace will be spanned by the ITOs {T 1 k } k .The reduced matrix element is independent of the vector label component k.Therefore, due to the uniqueness of the 1-irrep, the quantity j A ||T 1 ||j A will be uniquely associated with the irreducible subspace of B(H A ) that transforms under the 1-irrep.This implies that j A ||T 1 ||j A is independent on the choice of orthonormal ITO basis.Analogous relation holds for j B ||S 1 ||j B .In face, we can fairly easily determine what the constant factor j B S 1 j B j A T 1 j A is.For this we would again use Wigner-Eckart theorem together with the standard form for ITOs S 1 0 and T 1 0 to evaluate a particular matrix element.We then get where the above makes no assumption on the ITOs S 1 other than it forming an orthonormal basis for the 1-irrep component.Therefore, the ratio of the reduced matrix elements S 1 and T 1 is 2j A +1 2j B +1 .Now, in order to arrive at a closed form formula for f 1 (E L ), we need to combine the above with with binomial expansions for the Clebsch-Gordan coefficients.First notice that one of the terms in the expression for f 1 (E L ) is given by j B , j B ; L, s|j A , j B + s 2 = 2j A + 1 2j B + 1 Remark that for the coefficients to be non-zero we need that −L ≥ s ≥ L and j A − j B − s ≥ 0, where we recall that L takes one of the positive values in the set {|j A − j B |, |j A − j B | + 1, ..., j A + j B }. Therefore, we get f 1 (E L ) = j B (j B + 1)(2j A + 1) j A (j A + 1)(2j B + 1) where in the summation only the terms for which the two binomials exist contribute, i.e. j A − j B − s ≥ 0 (note that these correspond exactly to non-zero values of the relevant coefficients in the previous summation).Changing the dummy summation variable from s to w = s + L we obtain the alternative formulation: f 1 (E L ) = j B (j B + 1)(2j A + 1) j A (j A + 1)(2j B + 1) To compute the above, we make use of the following combinatorial property: The minimal value in turn will always be attained by L = j A + j B , which gives f 1 (E j A +j B ) = − j A j B + 1 j B (j B + 1)(2j A + 1) j A (j A + 1)(2j B + 1) .(C26) Note that for j B > j A there may exist an extremal channel for which the scaling coefficient will be less than −1.In other words, the spin polarisation can be effectively inverted in this case.
with the last inequality coming from • ∞ ≤ • 1 .Since p λ (E) forms a probability distribution over λ ∈ Λ we get the following bound on the deviation: (D21) Furthermore, we can bound the term M λ,k 1 for any λ and k.This follows from triangle inequality and submultiplicativity of the Schatten p-norms: However, with s(A) denoting the singular values of operator A. Since E λ † i E λ i ≥ 0 then it follows that E λ i 2 1 ≤ d A tr(E λ † i E λ i ).We also have that i tr(E λ † i E λ i ) = d A , as the Jamio lkowski states J (E λ ) satisfy tr(J (E λ )) = 1, or equivalently tr(E λ ( I d A )) = 1.Then we get the following upper bound on M λ,k : Therefore, we get the following upper bound on the deviation from a conservation law: where we recall that n is the number of generators and p 0 = p 0 (E).Combining the above with Eq. (D9) we finally obtain where analogously to the previous considerations we have introduced It is clear that f (λ) = 1 if and only if λ = 0.This is because for λ = 0 we deal with the identity channel and so f (λ) = 1; conversely, λ = 1 means that the J k operators are fixed points of the unital CPTP map, and so they commute with the Kraus operators, and this happens only for Kraus operators transforming as λ = 0.Moreover, without loss of generality, we may assume that |f (λ)| ≤ 1.This follows from a result of Ref. [37], which states that for unital trace-preserving channels the induced p-norm is contractive for all 1 < p ≤ ∞.That means that because E λ are unital CPTP maps due to the fact that we deal with an irrep system.Then, it follows that Since K arises from minimisation over all λ = 0, it is strictly greater than zero, leading to a non-trivial lower bound on the deviation.The coefficient K will be fixed for any given symmetry principle described by the representation U A of G. Now, according to Eq. ( 44) and using the decomposition from Eq. ( 88), unitarity can be expressed in terms of the probability distribution p λ as follows: where we have used that the channel E is unital.We can then bound it using Cauchy-Schwartz inequality in the following way: Equivalently, Combining the two relations results in ) Symmetric unitaries < l a t e x i t s h a 1 _ b a s e 6 4 = " D + a o 7 V y y I f h 4 V Q b 6 k c z 1 a 1 r d l w c = " > A A A C C X i c b V C 7 T g M x E P S F V z h e A U o a i w i J K r p L A 2 U E D W U Q 5 C H l o s j n b B I r t u 9 k 7 y F F U V o a f o W G A o R o + Q M 6 / g b n U U D C S C u N Z n b t 3 Y l T K S w G w b e X W 1 v f 2 N z K b / s 7 u 3 v 7 B 4 X D o 7 p NM s O h x h O Z m G b M L E i h o Y Y C J T R T A 0 z F E h r x 8 H r q N x 7 A W J H o e x y l 0 F a s r 0 V P c I Z O 6 h R o p B O h u 6 D R v x s p B W g E j y L q Z 1 o g M w J s p 1 A M S s E M d J W E C 1 I k C 1 Q 7 h a + o m / B M u S e 5 Z N a 2 w i D F 9 p g Z F F z C x I 8 y C y n j Q 9 a H l q O a K b D t 8 e y S C T 1 z S p f 2 E u N K I 5 2 p v y f G T F k 7 U r H r V A w H d t m b i v 9 5 r Q x 7 l + 2 x 0 G m G o P n 8 o 1 4 m K S Z 0 G g v t C g M c 5 c g R x o 1 w u 1 I + Y I Z x d O H 5 L o R w + e R V U i + X w q A U 3 p a L l a t F H H l y Q k 7 J O Q n J B a m Q G 1 I l N c L J I 3 k m r + T N e / J e v H f v Y 9 6 a 8 x Y z x + Q P v M 8 f a R C a I Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D + a o 7 V y y I f h 4 V Q b 6 k c z 1 a 1 r d l w c = " > A A A C C X i c b V C 7 T g M x E P S F V z h e A U o a i w i J K r p L A 2 U E D W U Q 5 C H l o s j n b B I r t u 9 k 7 y F F U V o a f o W G A o R o + Q M 6 / g b n U U D C S C u N Z n b t 3 Y l T K S w G w b e X W 1 v f 2 N z K b / s 7 u 3 v 7 B 4 X D o 7 p N M s O h x h O Z m G b M L E i h o Y Y C J T R T A 0 z F E h r x 8 H r q N x 7 A W J H o e x y l 0 F a s r 0 V P c I Z O 6 h R o p B O h u 6 D R v x s p B W g E j y L q Z 1 o g M w J s p 1 A M S s E M d J W E C 1 I k C 1 Q 7 h a + o m / B M u S e 5 Z N a 2 w i D F 9 p g Z F F z C x I 8 y C y n j Q 9 a H l q O a K b D t 8 e y S C T 1 z S p f 2 E u N K I 5 2 p v y f G T F k 7 U r H r V A w H d t m b i v 9 5 r Q x 7 l + 2 x 0 G m G o P n 8 o 1 4 m K S Z 0 G g v t C g M c 5 c g R x o 1 w u 1 I + Y I Z x d O H 5 L o R w + e R V U i + X w q A U 3 p a L l a t F H H l y Q k 7 J O Q n J B a m Q G 1 I l N c L J I 3 k m r + T N e / J e v H f v Y 9 6 a 8 x Y z x + Q P v M 8 f a R C a I Q = = < / l a te x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D + a o 7 V y y I f h 4 V Q b 6 k c z 1 a 1 r d l w c = " > A A A C C X i c b V C 7 T g M x E P S F V z h e A U o a i w i J K r p L A 2 U E D W U Q 5 C H l o s j n b B I r t u 9 k 7 y F F U V o a f o W GA o R o + Q M 6 / g b n U U D C S C u N Z n b t 3 Y l T K S w G w b e X W 1 v f 2 N z K b / s 7 u 3 v 7 B 4 X D o 7 p N M s O h x h O Z m G b M L E i h o Y Y C J T R T A 0 z F E h r x 8 H r q N x 7 A W J H o e x y l 0 F a s r 0 V P c I Z O 6 h R o p B O h u 6 D R v x s p B W g E j y L q Z 1 o g M w J s p 1 A M S s E M d J W E C 1 I k C 1 Q 7 h a + o m / B M u S e 5 Z N a 2 w i D F 9 p g Z F F z C x I 8 y C y n j Q 9 a H l q O a K b D t 8 e y S C T 1 z S p f 2 E u N K I 5 2 p v y f G T F k 7 U r H r V A w H d t m b i v 9 5 r Q x 7 l + 2 x 0 G m G o P n 8 o 1 4 m K S Z 0 G g v t C g M c 5 c g R x o 1 w u 1 I + Y I Z x d O H 5 L o R w + e R V U i + X w q A U 3 p a L l a t F H H l y Q k 7 J O Q n J B a m Q G 1 I l N c L J I 3 k m r + T N e / J e v H f v Y 9 6 a 8 x Y z x + Q P v M 8 f a R C a I Q = = </ l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D + a o 7 V y y I f h 4 V Q b 6 k c z 1 a 1 r d l w c = " > A A A C C X i c b V C 7 T g M x E P S F V z h e A U o a i w i J K r p L A 2 U E D W U Q 5 C H l o s j n b B I r t u 9 k 7 y F F U V o a f o W G A o R o + Q M 6 / g b n U U D C S C u N Z n b t 3 Y l T K S w G w b e X W 1 v f 2 N z K b / s 7 u 3 v 7 B 4 X D o 7 p N M s O h x h O Z m G b M L E i h o Y Y C J T R T A 0 z F E h r x 8 H r q N x 7 A W J H o e x y l 0 F a s r 0 V P c I Z O 6 h R o p B O h u 6 D R v x s p B W g E j y L q Z 1 o g M w J s p 1 A M S s E M d J W E C 1 I k C 1 Q 7 h a + o m / B M u S e 5 Z N a 2 w i D F 9 p g Z F F z C x I 8 y C y n j Q 9 a H l q O a K b D t 8 e y S C T 1 z S p f 2 E u N K I 5 2 p v y f G T F k 7 U r H r V A w H d t m b i v 9 5 r Q x 7 l + 2 x 0 G m G o P n 8 o 1 4 m K S Z 0 G g v t C g M c 5 c g R x o 1 w u 1 I + Y I Z x d O H 5 L o R w + e R V U i + X w q A U 3 p a L l a t F H H l y Q k 7 J O Q n J B a m Q G 1 I l N c L J I 3 k m r + T N e / J e v H f v Y 9 6 a 8 x Y z x + Q P v M 8 f a R C a I Q = = < / l a t e x i t > 4 r 5 7 c H h 0 f F I 6 P W t r m S k K L S q 5V N 2 I a O B M Q M s w w 6 G b K i B J x K E T j R s L v z M B p Z k U D 2 a a Q p i Q o W A D R o m x U r 9 U C Y R k I g Z h 3 M a I q C F g K o U G N W F i G A S u T E H l S d 0 v l b 2 q l w N v E n 9 F y m i F Z r / 0 F c S S Z o m 9 m n K i d c / 3 U h P O i D K M c p i 7 Q a Y h J X R M h t C z V J A E d D j L N 5r j K 6 v E e C C V P c L g X P 0 9 M S O J 1 t M k s s m E m J F e 9 x b i f 1 4 v M 4 O b c M Z E m h k Q d P n Q I O P Y S L y o B 8 d M A T V 8 a g m h i t m / Y m p 7 I d T Y E l 1 b g 4 r 5 7 c H h 0 f F I 6 P W t r m S k K L S q 5V N 2 I a O B M Q M s w w 6 G b K i B J x K E T j R s L v z M B p Z k U D 2 a a Q p i Q o W A D R o m x U r 9 U C Y R k I g Z h 3 M a I q C F g K o U G N W F i G A S u T E H l S d 0 v l b 2 q l w N v E n 9 F y m i F Z r / 0 F c S S Z o m 9 m n K i d c / 3 U h P O i D K M c p i 7 Q a Y h J X R M h t C z V J A E d D j L N 5r j K 6 v E e C C V P c L g X P 0 9 M S O J 1 t M k s s m E m J F e 9 x b i f 1 4 v M 4 O b c M Z E m h k Q d P n Q I O P Y S L y o B 8 d M A T V 8 a g m h i t m / Y m p 7 I d T Y E l 1 b g 4 r 5 7 c H h 0 f F I 6 P W t r m S k K L S q 5V N 2 I a O B M Q M s w w 6 G b K i B J x K E T j R s L v z M B p Z k U D 2 a a Q p i Q o W A D R o m x U r 9 U C Y R k I g Z h 3 M a I q C F g K o U G N W F i G A S u T E H l S d 0 v l b 2 q l w N v E n 9 F y m i F Z r / 0 F c S S Z o m 9 m n K i d c / 3 U h P O i D K M c p i 7 Q a Y h J X R M h t C z V J A E d D j L N 5r j K 6 v E e C C V P c L g X P 0 9 M S O J 1 t M k s s m E m J F e 9 x b i f 1 4 v M 4 O b c M Z E m h k Q d P n Q I O P Y S L y o B 8 d M A T V 8 a g m h i t m / Y m p 7 I d T Y E l 1 b g 4 r 5 7 c H h 0 f F I 6 P W t r m S k K L S q 5 V N 2 I a O B M Q M s w w 6 G b K i B J x K E T j R s L v z M B p Z k U D 2 a a Q p i Q o W A D R o m x U r 9 U C Y R k I g Z h 3 M a I q C F g K o U G N W F i G A S u T E H l S d 0 v l b 2 q l w N v E n 9 F y m i F Z r / 0 F c S S Z o m 9 m n K i d c / 3 U h P O i D K M c p i 7 Q a Y h J X R M h t C z V J A E d D j L N 5 r j K 6 v E e C C V P c L g X P 0 9 M S O J 1 t M k s s m E m J F e 9 x b i f 1 4 v M 4 O b c M Z E m h k Q d P n Q I O P Y S L y o B 8 d M A T V 8 a g m h i t m / Y m p 7 I d T Y E l 1 b g r + + 8 i Z p 1 6 q + V / X v a + X 6 7 a q O I r p A l 6 i C f H S N 6 u g O N V E L U f S I n t E r e n O e n B f n 3 f l Y R g v O a u Y c / Y H z + Q P j l p 2 s < / l a t e x i t > S y m m e t r ic o p e r a t io n s < l a t e x i t s h a 1 _ b a s e 6 4 = " q 9G Z S m 8 N 2 X i Q E J m r 7 N w c + o h D f / Y = " > A A A C B 3 i c b V C 7 S g N B F J 3 1 G e N r 1 V K Q w S B Y h d 0 0 W g Z t L C O a B y Q h z M 7 e T Y b M Y 5 m Z F U J I Z + O v 2 F g o Y u s v 2 P k 3 T j Y p N P H A h c M 5 9 8 7 c e 6 K U M 2 O D 4 N t b W V 1 b 3 9 g s b B W 3 d 3 b 3 9 v 2 D w 4 Z R m a Z Q p 4 o r 3 Y q I A c 4 k 1 C 2 z H F q p B i I i D s 1 o e D 3 1 m w + g D V P y 3 o 5 S 6 A r S l y x h l F g n 9 f y T j l R M x i B t 8 W 4 k B F j N K F Y p 6 N w 3 P b 8 U l I M c e J m E c 1 J C c 9 R 6 / l c n V j Q T 7 k H K i T H t M E h t d 0 y 0 Z Z T D p N j J D K S E D k k f 2 o 5 K I s B 0 x / k d E 3 z m l B g n S r u S F u f q 7 4 k x E c a M R O Q 6 B b E D s + h N x f + 8 d m a T y + 6 Y y T S z I O n s o y T j 2 C o 8 D Q X H T A O 1 f O Q I o Z q 5 X T E d E E 2 o d d E V X Q j h 4 s n L p F E p h 0 E 5 v K 2 U q l f z O A r o G J 2 i c x S i C 1 R F N 6 i G 6 o i i R / S MX t G b 9 + S 9 e O / e x 6 x 1 x Z v P H K E / 8 D 5 / A I 4 4 m b s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q 9 G Z S m 8 N 2 X i Q E J m r 7 N w c + o h D f / Y = " > A A A C B 3 i c b V C 7 S g N B F J 3 1 G e N r 1 V K Q w S B Y h d 0 0 W g Z t L C O a B y Q h z M 7 e T Y b M Y 5 m Z F U J I Z + O v 2 F g o Y u s v 2 P k 3 T j Y p N P H A h c M 5 9 8 7 c e 6 K U M 2 O D 4 N t b W V 1 b 3 9 g s b B W 3 d 3 b 3 9 v 2 D w 4 Z R m a Z Q p 4 o r 3 Y q I A c 4 k 1 C 2 z H F q p B i I i D s 1 o e D 3 1 m w + g D V P y 3 o 5 S 6 A r S l y x h l F g n 9 f y T j l R M x i B t 8 W 4 k B F j N K F Y p 6 N w 3 P b 8 U l I M c e J m E c 1 J C c 9 R 6 / l c n V j Q T 7 k H K i T H t M E h t d 0 y 0 Z Z T D p N j J D K S E D k k f 2 o 5 K I s B 0 x / k d E 3 z m l B g n S r u S F u f q 7 4 k x E c a M R O Q 6 B b E D s + h N x f + 8 d m a T y + 6 Y y T S z I O n s o y T j 2C o 8 D Q X H T A O 1 f O Q I o Z q 5 X T E d E E 2 o d d E V X Q j h 4 s n L p F E p h 0 E 5 v K 2 U q l f z O A r o G J 2 i c x S i C 1 R F N 6 i G 6 o i i R / S MX t G b 9 + S 9 e O / e x 6 x 1 x Z v P H K E / 8 D 5 / A I 4 4 m b s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q 9 G Z S m 8 N 2 X i Q E J m r 7 N w c + o h D f / Y = " > A A A C B 3 i c b V C 7 S g N B F J 3 1 G e N r 1 V K Q w S B Y h d 0 0 W g Z t L C O a B y Q h z M 7 e T Y b M Y 5 m Z F U J I Z + O v 2 F g o Y u s v 2 P k 3 T j Y p N P H A h c M 5 9 8 7 c e 6 K U M 2 O D 4 N t b W V 1 b 3 9 g s b B W 3 d 3 b 3 9 v 2 D w 4 Z R m a Z Q p 4 o r 3 Y q I A c 4 k 1 C 2 z H F q p B i I i D s 1 o e D 3 1 m w + g D V P y 3 o 5 S 6 A r S l y x h l F g n 9 f y T j l R M x i B t 8 W 4 k B F j N K F Y p 6 N w 3 P b 8 U l I M c e J m E c 1 J C c 9 R 6 / l c n V j Q T 7 k H K i T H t M E h t d 0 y 0 Z Z T D p N j J D K S E D k k f 2 o 5 K I s B 0 x / k d E 3 z m l B g n S r u S F u f q 7 4 k x E c a M R O Q 6 B b E D s + h N x f + 8 d m a T y + 6 Y y T S z I O n s o y T j 2 C o 8 DQ X H T A O 1 f O Q I o Z q 5 X T E d E E 2 o d d E V X Q j h 4 s n L p F E p h 0 E 5 v K 2 U q l f z O A r o G J 2 i c x S i C 1 R F N 6 i G 6 o i i R / S MX t G b 9 + S 9 e O / e x 6 x 1 x Z v P H K E / 8 D 5 / A I 4 4 m b s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " q 9 G Z S m 8 N 2 X i Q E J m r 7 N w c + o h D f / Y = " > A A A C B 3 i c b V C 7 S g N B F J 3 1 G e N r 1 V K Q w S B Y h d 0 0 W g Z t L C O a B y Q h z M 7 e T Y b M Y 5 m Z F U J I Z + O v 2 F g o Y u s v 2 P k 3 T j Y p N P H A h c M 5 9 8 7 c e 6 K U M 2 O D 4 N t b W V 1 b 3 9 g s b B W 3 d 3 b 3 9 v 2 D w 4 Z R m a Z Q p 4 o r 3 Y q I A c 4 k 1 C 2 z H F q p B i I i D s 1 o e D 3 1 m w + g D V P y 3 o 5 S 6 A r S l y x h l F g n 9 f y T j l R M x i B t 8 W 4 k B F j N K F Y p 6 N w 3 P b 8 U l I M c e J m E c 1

FIG. 1 .
FIG. 1. Disconnect between symmetries and conservation laws for open quantum dynamics.Every continuous symmetry of the closed unitary evolution implies a conserved charge, but under the same symmetry constraints quantum channels may change the expectation value of such charges.
n H O k I T Z J A P S Y p 0 T w 1 B B P J z K 2 I D L H E R J u 8 C i a E p Z e X S b N a c Z 2 K e 1 M t 1 S 7 n c e T h E I 7 g B F w 4 g x p c Q x 0 a Q C C F J 3 i B V + v R e r b e r P d Z a 8 6 a z x T h D 6 y P b 7 w j k 4 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 M u T Q L C W t C B G V 4 h q S k T C h C P w Z w E = " > A A A B / H i c b V D L S s N A F L 2 p r 1 p f 0 S 7 d D L a C G 0 v S j S 6 r b l x W t A 9 o Q 5 h M p + 3 Y y S T M T I Q Q 6 q + 4 c a G I W z / E n X / j 9 I F o 6 4 E L h 3 P u 5 d 5 7 g p g z p R 3 n y 8 q t r K 6 t b n H O k I T Z J A P S Y p 0 T w 1 B B P J z K 2 I D L H E R J u 8 C i a E p Z e X S b N a c Z 2 K e 1 M t 1 S 7 n c e T h E I 7 g B F w 4 g x p c Q x 0 a Q C C F J 3 i B V + v R e r b e r P d Z a 8 6 a z x T h D 6 y P b 7 w j k 4 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 M u T Q L C W t C B G V 4 h q S k T C h C P w Z w E = " > A A A B / H i c b V D L S s N A F L 2 p r 1 p f 0 S 7 d D L a C G 0 v S j S 6 r b l x W t A 9 o Q 5 h M p + 3 Y y S T M T I Q Q 6 q + 4 c a G I W z / E n X / j 9 I F o 6 4 E L h 3 P u 5 d 5 7 g p g z p R 3 n y 8 q t r K 6 t b n H O k I T Z J A P S Y p 0 T w 1 B B P J z K 2 I D L H E R J u 8 C i a E p Z e X S b N a c Z 2 K e 1 M t 1 S 7 n c e T h E I 7 g B F w 4 g x p c Q x 0 a Q C C F J 3 i B V + v R e r b e r P d Z a 8 6 a z x T h D 6 y P b 7 w j k 4 Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 M u T Q L C W t C B G V 4 h q S k T C h C P w Z w E = " > A A A B / H i c b V D L S s N A F L 2 p r 1 p f 0 S 7 d D L a C G 0 v S j S 6 r b l x W t A 9 o Q 5 h M p + 3 Y y S T M T I Q Q 6 q + 4 c a G I W z / E n X / j 9 I F o 6 4 E L h 3 P u 5 d 5 7 g p g z p R 3 n y 8 q t r K 6 t b n H O k I T Z J A P S Y p 0 T w 1 B B P J z K 2 I D L H E R J u 8 C i a E p Z e X S b N a c Z 2 K e 1 M t 1 S 7 n c e T h E I 7 g B F w 4 g x p c Q x 0 a Q C C F J 3 i B V + v R e r b e r P d Z a 8 6 a z x T h D 6 y P b 7 w j k 4 Y = < / l a t e x i t > Spin -j B system < l a t e x i t s h a 1 _ b a s e 6 4 = " / B 6 c I D F F c h l 2 b v y u 0A X u J 4 4 X U 3 E = " > A A A B / H i c d V D L S g M x F M 3 U V 6 2 v 0 S 7 d B F v B j S V T q L a 7 U j c u K 9 o H t K V k 0 r S N T T J D k h G G o f 6 K G x e K u P V D 3 P k 3 p g 9 B R Q 9 c O J x z L / f e 4 4 e c a Y P Q h 5 N a W V 1 b 3 0 h v Z r a 2 d 3 b 3 3 P 2 D p g 4 i R W i D B D x Q b R 9 r y p m k D c M M p + 1 Q U S x 8 T l v + 5 G L m t + 6 o 0 i y Q N y Y O a U / g k W R D R r C x U t / N X o d M w l O Y v + 3 X 8 lD H 2 l D R d 3 O o U E J e 5 c y D q I D m s K R Y Q p U y g t 5 S y Y E l 6 n 3 3 v T s I S C S o N I R j r T s e C k 0 v w c o w w u k 0 0 4 0 0D T G Z 4 B H t W C q x o L q X z I + f w m O r D O A w U L a k g X P 1 + 0 S C h d a x 8 G 2 n w G a s f 3 s z 8 S + v E 5 l h u Z c w G U a G S r J Y N I w 4 N A G c J Q E H T F F i e G w J J o r Z W y E Z Y 4 W J s X l l b A h f n 8 L / S b N Y 8 F D B u y r m q r V l H G l w C I 7 A C f D A O a i C S 1 A H D U B A D B 7 A E 3 h 2 7 p 1 H 5 8 V 5 X b S m n O V M F v y A 8 / Y J / r u T t A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / B 6 c I D F F c h l 2 b v y u 0 A X u J 4 4 X U 3 E = " > A A A B / H i c d V D L S g M x F M 3 U V 6 2 v 0 S 7 d B F v B j S V T q L a 7 U j c u K 9 o H t K V k 0 r S N T T J D k h G G o f 6 K G x e K u P V D 3 P k 3 p g 9 B R Q 9 c O J x z L / f e 4 4 e c a Y P Q h 5 N a W V 1 b 3 0 h v Z r a 2 d 3 b 3 3 P 2 D p g 4 i R W i D B D x Q b R 9 r y p m k D c M M p + 1 Q U S x 8 T l v + 5 G L m t + 6 o 0 i y Q N y Y O a U / g k W R D R r C x U t / N X o d M w l O Y v + 3 X 8 lD H 2 l D R d 3 O o U E J e 5 c y D q I D m s K R Y Q p U y g t 5 S y Y E l 6 n 3 3 v T s I S C S o N I R j r T s e C k 0 v w c o w w u k 0 0 4 0 0 D T G Z 4 B H t W C q x o L q X z I + f w m O r D O A w U L a k g X P 1 + 0 S C h d a x 8 G 2 n w G a s f 3 s z 8S + v E 5 l h u Z c w G U a G S r J Y N I w 4 N A G c J Q E H T F F i e G w J J o r Z W y E Z Y 4 W J s X l l b A h f n 8 L / S b N Y 8 F D B u y r m q r V l H G l w C I 7 A C f D A O a i C S 1 A H D U B A D B 7 A E 3 h 2 7 p 1 H 5 8 V 5 X b S m n O V M F v y A 8 / Y J / r u T t A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / B 6 c I D F F c h l 2 b v y u 0 A X u J 4 4 X U 3 E = " >A A A B / H i c d V D L S g M x F M 3 U V 6 2 v 0 S 7 d B F v B j S V T q L a 7 U j c u K 9 o H t K V k 0 r S N T T J D k h G G o f 6 K G x e K u P V D 3 P k 3 p g 9 B R Q 9 c O J x z L / f e 4 4 e c a Y P Q h 5 N a W V 1 b 3 0 h v Z r a 2 d 3 b 3 3 P 2 D p g 4 i R W i D B D x Q b R 9 r y p m k D c M M p + 1 Q U S x 8 T l v + 5 G L m t + 6 o 0 i y Q N y Y O a U / g k W R D R r C x U t / N X o d M w l O Y v + 3 X 8 l D H 2 l D R d 3 O o U E Je 5 c y D q I D m s K R Y Q p U y g t 5 S y Y E l 6 n 3 3 v T s I S C S o N I R j r T s e C k 0 v w c o w w u k 0 0 4 0 0 D T G Z 4 B H t W C q x o L q X z I + f w m O r D O A w U L a k g X P 1 + 0 S C h d a x 8 G 2 n w G a s f 3 s z 8 S + v E 5 l h u Z c w G U a G S r J Y N I w 4 N A G c J Q E H T F F i e G w J J o r Z W y E Z Y 4 W J s X l l b A h f n 8 L / S b N Y 8 F D B u y r m q r V l H G l w C I 7 A C f D A O a i C S 1 A H D U B A D B 7 A E 3 h 2 7 p 1 H 5 8 V 5 X b S m n O V M F v y A 8 / Y J / r u T t A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / B 6 c I D F F c h l 2 b v y u 0 A X u J 4 4 X U 3 E = " > A A A B / H i c d V D L S g M x F M 3 U V 6 2 v 0 S 7 d B F v B j S V T q L a 7 U j c u K 9 o H t K V k 0 r S N T T J D k h G G o f 6 K G x e K u P V D 3 P k 3 p g 9 B R Q 9 c O J x z L / f e 4 4 e c a Y P Q h 5 N a W V 1 b 3 0 h v Z r a 2 d 3 b 3 3 P 2 D p g 4 i R W i D B D x Q b R 9 r y p m k D c M M p + 1 Q U S x 8 T l v + 5 G L m t + 6 o 0 i y Q N y Y O a U / g k W R D R r C x U t / N X o d M w l O Y v + 3 X 8 l D H 2 l D R d 3 O o U E J e 5 c y D q I D m s K R Y Q p U y g t 5 S y Y E l 6 n 3 3 v T s I S C S o N I R j r T s e C k 0 v w c o w w u k 0 0 4 0 0 r V K / y u s o w g m c w j l 4 c A F 1 u I E G N I G B g m d 4 h T f n y X l x 3 p 2 P R b T g 5 D P H 8 A f O 5 w + s K p K b < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Y q 6 x K 6 l k k a o / W x + 9 Z S c W W V A w l j Z J w 2 Z q 7 8 n M h p p P Y k C m 4 y o G e l l b y b + 5 3 V T E 1 7 6 G Z d J a l C y x a I w F c T E Z F Y B G X C F z I i J J Z Q p b m 8 l b E Q V Z c b W U L I l e M t f X i W t W t V z q 9 5 t r V K / y u s o w g m c w j l 4 c A F 1 u I E G N I G B g m d 4 h T f n y X l x 3 p 2 P R b T g 5 D P H 8 A f O 5 w + s K p K b < / l a t e x i t > Spin amplification < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 7 x Y P g K m P w 5 j z U 9 m 8 b R l t L z H + b o = " > A A A B + 3 i c b V D L S g N B E O z 1 G e M r x q O X w S B 4 C r u 5 6 D H o x W N E 8 4 B k C b O T 2 W T I P J a Z W T E s + R U v H h T x 6 o 9 4 8 2 + c J A t q Y k F D U d V N d 1 e U c G a s 7 3 9 5 a + s b m 1 v b h Z 3 i 7 t 7 + w W H p q N w y K t W E N o n i S n c i b C h n k j Y t s 5 x 2 E k 2 x i D h t R + P r m d 9 + o N o w J e / t J K G h w E P J Y k a w d V K / V L 5 L m E R Y u F 0 / Y s W v + n O g V R L k p A I 5 G v 3 S Z 2 + g S C q o t I R j Y 7 q B n 9 g w w 9 o y w u m 0 2 E s N T T A Z 4 y H t O i q x o C b M 5 r d P 0 Z l T B i h W 2 p W 0 a K 7 + n s i w M G Y i I t c p s B 2 Z Z W 8 m / u d 1 U x t f h h m T S W q p J I t F c c q R V W g W B B o w T Y n l E 0 c w 0 c z d i s g I a 0 y s i 6 v o Q g i W X 1 4 l r V o 1 8 K v B b a 1 S v 8 r j K M A J n M I 5 B H A B d b i B B j S B w C M 8 w Q u 8 e l P v 2 X v z 3 h e t a 1 4 + c w x / 4 H 1 8 A x q G l H c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 7 x Y P g K m P w 5 j z U 9 m 8 b R l t L z H + b o = " > A A A B + 3 i c b V D L S g N B E O z 1 G e M r x q O X w S B 4 C r u 5 6 D H o x W N E 8 4 B k C b O T 2 W T I P J a Z W T E s + R U v H h T x 6 o 9 4 8 2 + c J A t q Y k F D U d V N d 1 e U c G a s 7 3 9 5 a + s b m 1 v b h Z 3 i 7 t 7 + w W H p q N w y K t W E N o n i S n c i b C h n k j Y t s 5 x 2 E k 2 x i D h t R + P r m d 9 + o N o w J e / t J K G h w E P J Y k a w d V K / V L 5 L m E R Y u F 0 / Y s W v + n O g V R L k p A I 5 G v 3 S Z 2 + g S C q o t I R j Y 7 q B n 9 g w w 9 o y w u m 0 2 E s N T T A Z 4 y H t O i q x o C b M 5 r d P 0 Z l T B i h W 2 p W 0 a K 7 + n s i w M G Y i I t c p s B 2 Z Z W 8 m / u d 1 U x t f h h m T S W q p J I t F c c q R V W g W B B o w T Y n l E 0 c w 0 c z d i s g I a 0 y s i 6 v o Q g i W X 1 4 l r V o 1 8 K v B b a 1 S v 8 r j K M A J n M I 5 B H A B d b i B B j S B w C M 8 w Q u 8 e l P v 2 X v z 3 h e t a 1 4 + c w x / 4 H 1 8 A x q G l H c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 7 x Y P g K m P w 5 j z U 9 m 8 b R lt L z H + b o = " > A A A B + 3 i c b V D L S g N B E O z 1 G e M r x q O X w S B 4 C r u 5 6 D H o x W N E 8 4 B k C b O T 2 W T I P J a Z W T E s + R U v H h T x 6 o 9 4 8 2 + c J A t q Y k F D U d V N d 1 e U c G a s 7 3 9 5 a + s b m 1 v b h Z 3 i 7 t 7 + w W H p q N w y K t W E N o n i S n c i b C h n k j Y t s 5 x 2 E k 2 x i D h t R + P r m d 9 + o N o w J e / t J K G h w E P J Y k a w d V K / V L 5 L m E R Y u F 0 / Y s W v + n O g V R L k p A I 5 G v 3 S Z 2 + g S C q o t I R j Y 7 q B n 9 g w w 9 o y w u m 0 2 E s N T T A Z 4 y H t O i q x o C b M 5 r d P 0 Z l T B i h W 2 p W 0 a K 7 + n s i w M G Y i I t c p s B 2 Z Z W 8 m / u d 1 U x t f h h m T S W q p J I t F c c q R V W g W B B o w T Y n l E 0 c w 0 c z d i s g I a 0 y s i 6 v o Q g i W X 1 4 l r V o 1 8 K v Bb a 1 S v 8 r j K M A J n M I 5 B H A B d b i B B j S B w C M 8 w Q u 8 e l P v 2 X v z 3 h e t a 1 4 + c w x / 4 H 1 8 A x q G l H c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 7 x Y P g K m P w 5 j z U 9 m 8 b R l t L z H + b o = " > A A A B + 3 i c b V D L S g N B E O z 1 G e M r x q O X w S B 4 C r u 5 6 D H o x W N E 8 4 B k C b O T 2 W T I P J a Z W T E s + R U v H h T x 6 o 9 4 8 2 + c J A t q Y k F D U d V N d 1 e U c G a s 7 3 9 5 a + s b m 1 v b h Z 3 i 7 t 7 + w W H p q N w y K t W E N o n i S n c i b C h n k j Y t s 5 x 2 E k 2 x i D h t R + P r m d 9 + o N o w J e / t J K G h w E P J Y k a w d V K / V L 5 L m E R Y u F 0 / Y s W v + n O g V R L k p A I 5 G v 3 S Z 2 + g S C q o t I R j Y 7 q B n 9 g w w 9 o y w u m 0 2 E s N T T A Z 4 y H t O i q x o C b M 5 r d P 0 Z l T B i h W 2 p W 0 a K 7 + n s i w M G Y i I t c p s B 2 Z Z W 8 m / u d 1 U x t f h h m T S W q p J I t F c c q R V W g W B B o w T Y n l E 0 c w 0 c z d i s g I a 0 y s i 6 v o Q g i W X 1 4 l r V o 1 8 K v B b a 1 S v 8 r j K M A J n M I 5 B H A B d b i B B j S B w C M 8 w Q u 8 e l P v 2 X v z 3 h e t a 1 4 + c w x / 4 H 1 8 A x q G l H c = < / l a t e x i t > P(⇢) < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 L J U 7 R 4 3 0 y b h E C D 8 n u z b / o A n 9 3 Y = " > A A A B + X i c b V D L S g M x F L 1 T X 7 W + R l 2 6 C R a h b s q M C L o s u n F Z w T 6 g U 0 o m z b S h m W R I M o U y 9 E / c u F D E r X / i z r 8 x 0 8 5 C W w 8 E D u f c y z 0 5 Y c K Z N p 7 3 7 Z Q 2 N r e 2 d 8 q 7 l b 3 9 g 8 M j 9 / i k r W W q C G 0 R y a X q h l h T z g R t G W Y 4 7 S a K 4 j j k t B N O 7 n O / M 6 V K M y m e z C y h / R i P B I s Y w c Z K A 9 c N Y m z G Y Z Q 1 5 7 V A j e X l w K 1 6 d W 8 B t E 7 8 g l S h Q H P g f g V D S d K Y C k M 4 1 r r n e 4 n p Z 1 g Z R j i d V 4 J U 0 w S T C R 7 R n q U C x 1 T 3 s 0 X y O b q w y h B F U t k n D F q o v z c y H G s 9 i 0 M 7 m e f U q 1 4 u / u f 1 U h P d 9 j M m k t R Q Q Z a H o p Q j I 1 F e A x o y R Y n h M 0 s w U c x m R W S M F S b G l l W x J f i r X 1 4 n 7 a u 6 7 9 X 9 x + t q 4 6 6 o o w x n c A 4 1 8 O E G G v A A T W g B g S k 8 w y u 8 O Z n z 4 r w 7 H 8 v R k l P s n M I f O J 8 / L J u T V w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 L J U 7 R 4 3 0 y b h E C D 8 n u z b / o A n 9 3 Y = " > A A A B + X i c b V D L S g M x F L 1 T X 7 W + R l 2 6 C R a h b s q M C L o s u n F Z w T 6 g U 0 o m z b S h m W R I M o U y 9 E / c u F D E r X / i z r 8 x 0 8 5 C W w 8 E D u f c y z 0 5 Y c K Z N p 7 3 7 Z Q 2 N r e 2 d 8 q 7 l b 3 9 g 8 M j 9/ i k r W W q C G 0 R y a X q h l h T z g R t G W Y 4 7 S a K 4 j j k t B N O 7 n O / M 6 V K M y m e z C y h / R i P B I s Y w c Z K A 9 c N Y m z G Y Z Q 1 5 7 V A j e X l w K 1 6 d W 8 B t E 7 8 g l S h Q H P g f g V D S d K Y C k M 4 1 r r n e 4 n p Z 1 g Z R j i d V 4 J U 0 w S T C R 7 R n q U C x 1 T 3 s 0 X y O b q w y h B F U t k n D F q o v z c y H G s 9 i 0 M 7 m e f U q 1 4 u / u f 1 U h P d 9 j M m k t R Q Q Z a H o p Q j I 1 F e A x o y R Y n h M 0 s w U c x m R W S M F S b G l l W x J f ir X 1 4 n 7 a u 6 7 9 X 9 x + t q 4 6 6 o o w x n c A 4 1 8 O E G G v A A T W g B g S k 8 w y u 8 O Z n z 4 r w 7 H 8 v R k l P s n M I f O J 8 / L J u T V w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 L J U 7 R 4 3 0 y b h E C D 8 n u z b / o A n 9 3 Y = " > A A A B + X i c b V D L S g M x F L 1 T X 7 W + R l 2 6 C R a h b s q M C L o s u n F Z w T 6 g U 0 o m z b S h m W R I M o U y 9 E / c u F D E r X / i z r 8 x 0 8 5 C W w 8 E D u f c y z 0 5 Y c K Z N p 7 3 7 Z Q 2 N r e 2 d 8 q 7 l b 3 9 g 8 M j 9 / i k r W W q C G 0 R y a X q h l h T z g R t G W Y 4 7 S a K 4 j j k t B N O 7 n O / M 6 V K M y m e z C y h / R i P B I s Y w c Z K A 9 c N Y m z G Y Z Q 1 5 7 V A j e X l w K 1 6 d W 8 B t E 7 8 g l S h Q H P g f g V D S d K Y C k M 4 1 r r n e 4 n p Z 1 g Z R j i d V 4 J U 0 w S T C R 7 R n q U C x 1 T 3 s 0 X y O b q w y h B F U t k n D F q o v z c y H G s 9 i 0 M 7 m e f U q 1 4 u / u f 1 U h P d 9 j M m k t R Q Q Z a H o p Q j I 1 F e A x o y R Y n h M 0 s w U c x m R W S M F S b G l l W x J f i r X 1 4 n 7 a u 6 7 9 X 9 x + t q 4 6 6 o o w x n c A 4 1 8 O E G G v A A T W g B g S k 8 w y u 8 O Z n z 4 r w 7 H 8 v R k l P s n M I f O J 8 / L J u T V w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 L J U 7 R 4 3 0 y b h E C D 8 n u z b / o A n 9 3 Y = " > A A A B + X i c b V D L S g M x F L 1 T X 7 W + R l 2 6 C R a h b s q M C L o s u n F Z w T 6 g U 0 o m z b S h m W R I M o U y 9 E / c u F D E r X / i z r 8 x 0 8 5 C W w 8 E D u f c y z 0 5 Y c K Z N p 7 3 7 Z Q 2 N r e 2 d 8 q 7 l b 3 9 g 8 M j 9 / i k r W W q C G 0 R y a X q h l h T z g R t G W Y 4 7 S a K 4 j j k t B N O 7 n O / M 6 V K M y m e z C y h / R i P B I s Y w c Z K A 9 c N Y m z G Y Z Q 1 5 7 V A j e X l w K 1 6 d W 8 B t E 7 8 g l S h Q H P g f g V D S d K Y C k M 4 1 r r n e 4 n p Z 1 g Z R j i d V 4 J U 0 w S T C R 7 R n q U C x 1 T 3 s 0 X y O b q w y h B F U t k n D F q o v z c y H G s 9 i 0 M 7 m e f U q 1 4 u / u f 1 U h P d 9 j M m k t R Q Q Z a H o p Q j I 1 F e A x o y R Y n h M 0 s w U c x m R W S M F S b G l l W x J f i r X 1 4 n 7 a u 6 7 9 X 9 x + t q 4 6 6 o o w x n c A 4 1 8 O E G G v A A T W g B g S k 8 w y u 8 O Z n z 4 r w 7 H 8 v R k l P s n M I f O J 8 / L J u T V w = = < / l a t e x i t >  + P(⇢) < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 f k + S K 8 H z C 7 i f r 0 U F Y 1 G f P l S 9 G 0 = " > A A A C A 3 i c b V D L S s N A F J 3 4 r P U V d a e b Y B E q Q k l E 0 G X R j c s K 9 g F N C J P p p B 0 6 m R l m J k I J A T f + i h s X i r j 1 J 9 z 5 N 0 7 a L L T 1 w I X D O f d y 7 7 x I O u c N z 2 1 4 d x e 1 5 n U Z R w U c g W N Q B x 6 4 B E 1 w C 1 q g D R B 4 B M / g F b x Z T 9 a L 9 W 5 9 z F q X r H L m A P y B 9 f k D 4 V G X p g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 f k + S K 8 H z C 7 i f r 0 U F Y 1 G f P l S 9 G 0 = " > A A A C A 3 i c b V D L S s N A F J 3 4 r P U V d a e b Y B E q Q k l E 0 G X R j c s K 9 g F N C J P p p B 0 6 m R l m J k I J A T f + i h s X i r j 1 J 9 z 5 N 0 7 a L L T 1 w I X D O f d y 7 7 x I O u c N z 2 1 4 d x e 1 5 n U Z R w U c g W N Q B x 6 4 B E 1 w C 1 q g D R B 4 B M / g F b x Z T 9 a L 9 W 5 9 z F q X r H L m A P y B 9 f k D 4 V G X p g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 f k + S K 8 H z C 7 i f r 0 U F Y 1 G f P l S 9 G 0 = " > A A A C A 3 i c b V D L S s N A F J 3 4 r P U V d a e b Y B E q Q k l E 0 G X R j c s K 9 g F N C J P p p B 0 6 m R l m J k I J A T f + i h s X i r j 1 J 9 z 5 N 0 7 a L L T 1 w I X D O f d y 7 7 x I O u c N z 2 1 4 d x e 1 5 n U Z R w U c g W N Q B x 6 4 B E 1 w C 1 q g D R B 4 B M / g F b x Z T 9 a L 9 W 5 9 z F q X r H L m A P y B 9 f k D 4 V G X p g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 f k + S K 8 H z C 7 i f r 0 U F Y 1 G f P l S 9 G 0 = " > A A A C A 3 i c b V D L S s N A F J 3 4 r P U V d a e b Y B E q Q k l E 0 G X R j c s K 9 g F N C J P p p B 0 6 m R l m J k I J A T f + i h s X i r j 1 J 9 z 5 N 0 7 a L L T 1 w I X D O f d y 7

FIG. 3 .
FIG. 3. Robustness of Noether's principle & tradeoff relations.A qualitative description of trade-off relations between deviation from conservation laws and level of decoherence under the dynamics of a symmetric channel.While the red upper bound exists for all symmetries described by connected Lie groups, the lower bound is present when quantum systems have multiplicity-free decompositions.
C. Conservation laws vs decoherence: Quantitative trade-off relations

2 A 2 B
where V is a unitary operator.The Liouville representation of X ∈ B(H A ) is defined by a unique column vector |X ∈ C d 2 (as opposed to vectors in H S denoted by |• ) with entries given by the inner product tr(T † k X), where {T k } d k=1 is a fixed orthonormal basis of B(H A ).By analogously denoting a fixed orthonormal basis of B(H B ) by {S k } d k=1 , the Liouville representation of the superoperator E : B(H A ) → B(H B ) is a d 2 B by d 2 A matrix L(E) defined uniquely via the relation: and the summation is performed only over the indices m, n for which E m B − λ and E n B − λ correspond to valid energies of H A .

FIG. 4 .
FIG. 4.Trade-off between the deviation from angular momentum conservation, ∆(E), and unitarity u(E) for SU(2)-irreducibly-covariant channels.The middle blue line in the top panel (spin-1/2 system) and the blue dots in the bottom panel (spin-1 system) represent [∆(E), u(E)] pairs realised by covariant channels.The top red and bottom green curves give the upper and lower bounds specialised to the case of SU(2) symmetry.
FIG. 5.Trade-off between the deviation from energy conservation, ∆(E), and unitarity u(E) for U(1)covariant channels Each blue dot represents [∆(E), u(E)] pair for a qubit channel with fixed P E and optimal unitarity (the two parameters defining P E , 0 ≤ P E 00 , P E 11 ≤ 1, are taken as points from the lattice [0, 1] × [0, 1] with lattice constant 0.02).The orange solid line is the upper bound from Eq. (190).

j
B , n |E L (T λ k )|j B , m = m,n j A , n|T λ k |j A , m j B , n |E L (|j A , n j A , m|)|j B , m .(C2) −j A j A , n|T λ k |j A , m j B , n |E L (|j A , n j A , m|)|j B , m j B , n |S λ k |j B , m = j A m,n=−j A L s=−L j A , n|T λ k |j A , m j B , n |S λ k |j B , m j B , m − s; L, s|j A , m j B , n − s; L, s|j A , n δ n ,n−s δ m ,m−s = L s=−L j A , n + s|T λ k |j A , m + s j B , n |S λ k |j B , m j B , m ; L, s|j A , m + s j B , n ; L, s|j A , n + s .(C3)To simplify the above expression further we can employ the Wigner-Eckart theorem, which states that the matrix elements of an irreducible tensor operators depend on the vector component labels only trough the Clebsch-Gordan coefficients.In particular:j B , n |S λ k |j B , m = j B , m ; λ, k|j B , n j B ||S λ ||j B , (C4)where j B ||S λ ||j B is the reduced matrix element which is independent of n , m or k.We can also write down Wigner-Eckart for the T λ k irreducible operator.This leads to the following form for the vector of coefficients for the extremal channel labelled by L:f λ (E L ) = j A ||T λ ||j A j B ||S λ ||j B L s=−L j A ,m + s; λ, k|j A , n + s j B , m ; λ, k|j B , n j B , m ; L, s|j A , m + s j B , n ; L, s|j A , n + s .(C5) + s; 1, 0|j, j + s j, j; 1, 0|j, j = j + s j (C9) and j, j; L, s|j, j+ s 2 = (2j + 1)!(2j + s)!(L − s)! (2j − L)!(L + 2j + 1)!(L + s)!(−s)!(C10)is non-zero for s ≤ 0. As a result, we havef 1 (E L ) = (2j + 1)! (2j − L)!(L + 2j + 1)! L s=0 (j − s)(2j − s)!(L + s)! j(L − s)!s! .(C11)

2 .
a + b + c)! a!(1 + b + c)! = 1 + a + b + c a , a + b + c)!(1 + a + b) (a − 1)!(2 + b + c)! = 1 + a + b + c a a(1 + a + b) 2 + b + c (C21)for a = 0 and c ≥ a (if a = 0 the latter sum clearly becomes zero).Now, employing this we obtainf 1 (E L ) = j B (j B + 1)(2j A + 1) j A (j A + 1)(2j B + 1) j B − L j B + (j A − j B + L)(1 + j A + j B − L) 2j B (1 + j B ) ,(C22)and after some simplification we arrive atf 1 (E L ) = j B (j B + 1)(2j A + 1) j A (j A + 1)(2j B + 1) j A (j A + 1) + j B (j B + 1) − L(L + 1) 2j B (j B + 1) .(C23)For different extremal channels with L between |j B − j A | and j A + j B the maximal value is attained for the closest valid value of L to j A −j B +1 This maximal value is attained for L = |j A − j B |.We then have two cases.If j A ≥ j B thenf 1 (E |j A −j B | ) = j B (j A + 1)(2j A + 1) j A (j B + 1)(2j B + 1) ,(C24)and if j A ≤ j B then f 1 (E |j A −j B | ) = j A (j B + 1)(2j A + 1) j B (j A + 1)(2j B + 1) .(C25) evolves the system in time by t.The representation of the group on H B is defined in an analogous way with the Hamiltonian H B .Recall that, by Noether's theorem, closed unitary dynamics symmetric under timetranslations, generated by H A , conserves energy represented by Hamiltonian H A .
Theorem 11.Let G be a connected compact Lie group with unitary representations U A and U B acting on Hilbert spaces H A and H B , and generated by traceless generators {J k A } n k=1 and {J k B } n k=1 .For every G-covariant quantum channel E : B(H A ) → B(H B ) with d A ≥ d B the following holds: SW AP B ∈ B(H B ⊗ H B ) is the swap operator on system B. One can also show that tr(J[E] 2 ) = tr(SW AP B E ⊗ E(SW AP A ) by expanding in terms of basis for A and B. To check directly denote by |e m d B m=1 an orthonormal basis for system B. We get that tr(SW AP B E ⊗ E(SW AP A j B + s; 1, 0|j A , j B + s j B , j B ; 1, 0|j B , j B j B , j B ; L, s|j A , j B + s 2 .