Beyond i.i.d. in the Resource Theory of Asymmetry: An Information-Spectrum Approach for Quantum Fisher Information

Energetic coherence is indispensable for various operations, including precise measurement of time and acceleration of quantum manipulations. Since energetic coherence is fragile, it is essential to understand the limits in distillation and dilution to restore damage. The resource theory of asymmetry (RTA) provides a rigorous framework to investigate energetic coherence as a resource to break time-translation symmetry. Recently, in the i.i.d. regime where identical copies of a state are converted into identical copies of another state, it has been shown that the convertibility of energetic coherence is governed by a standard measure of energetic coherence, called the quantum Fisher information (QFI). This fact means that QFI in the theory of energetic coherence takes the place of entropy in thermodynamics and entanglement entropy in entanglement theory. However, distillation and dilution in realistic situations take place in regimes beyond i.i.d., where quantum states often have complex correlations. Unlike entanglement theory, the conversion theory of energetic coherence in pure states in the non-i.i.d. regime has been an open problem. In this Letter, we solve this problem by introducing a new technique: an information-spectrum method for QFI. Two fundamental quantities, coherence cost and distillable coherence, are shown to be equal to the spectral QFI rates for arbitrary sequences of pure states. As a consequence, we find that both entanglement theory and RTA in the non-i.i.d. regime are understood in the information-spectrum method, while they are based on different quantities, i.e., entropy and QFI, respectively.

Energetic coherence and other fundamental properties of quantum systems are better understood by treating them as resources for quantum tasks.Quantum resource theories (QRTs) provide a versatile framework for analyzing seemingly unrelated resources with different origins, including entanglement [22], athermality [23,24], and energetic coherence [5,[25][26][27].Unexpected similarities arise in different branches of QRTs [28], leading to a unified understanding of the underlying laws.Since valuable resources are often fragile, it is fundamental to develop a theoretical understanding of the distillation and dilution of resources to restore their damage.Here, distillation is the operation of extracting as much resource as possible from a given state, and dilution is the opposite (Fig. 1).
Revealing the limits of distillation and dilution of energetic coherence is of great importance when assembling multiple inaccurate clocks into an accurate clock [5].It

Dilution Distilation
FIG. 1. Schematic picture of dilution and distillation.In dilution, a given state or a given sequence of states (depicted as light blue liquid) is generated by consuming as little resource (depicted as dark blue liquid) as possible.In distillation, as much resource as possible is extracted from a given state or a given sequence of states.
has been studied [5,27] in the resource theory of asymmetry (RTA), a branch of QRTs that analyzes symmetries and conservation laws [12, 14-16, 25-27, 29-32].In the independent and identically distributed (i.i.d.) regime where identical copies of a pure state are converted to identical copies of another pure state, the conversion rate is shown to be given by the ratio of their quantum Fisher information (QFI), a quantifier of energetic coherence [27].In other words, QFI is in the position of entropy in the second law of thermodynamics.The same thermodynamic structure is known to exist in the i.i.d.regime for entanglement entropy in entanglement theory [33].
Towards practical applications, it is essential to extend the conversion theory in the i.i.d.regime to noni.i.d.setting because a realistic resource often has complex correlations while an i.i.d.resource state has no arXiv:2204.08439v6[quant-ph] 18 Nov 2023 correlation.In entanglement theory [34,35] and quantum thermodynamics [36][37][38], conversion theories in the non-i.i.d.regime have been established.However, the counterpart in RTA remains elusive.
The obstacle to analyzing non-i.i.d.regime in RTA is the limitation of the traditional information-spectrum method.This method gives a universal way of dealing with entropy-related problems for general states with arbitrary correlations in classical and quantum information theory, e.g., source coding, channel coding and hypothesis testing [39][40][41][42][43]. Furthermore, entanglement theory and quantum thermodynamics in the non-i.i.d.regime [34][35][36][37][38]44] are established with the information-spectrum method since they are based on entropy.However, a central measure in converting energetic coherence is QFI, which is quite different from entropy.Therefore, the non-i.i.d.theory in RTA has been out of the scope of the information-spectrum method.
In this Letter, we establish the conversion theory of energetic coherence in non-i.i.d.pure states by constructing an information-spectrum approach for QFI.The key ingredients we introduce here are the followings: the spectral sup-and inf-QFI rates, the max-and min-QFIs, and asymmetric majorization.All of them clarify the correspondence in the conversion theories of entanglement and energetic coherence in the non-i.i.d.regime, which are characterized by entropies and QFIs, respectively.First, we prove a general formula for the coherence cost and the distillable coherence, i.e., the optimal conversion rates of a sequence of arbitrary pure states from and to a reference state.Concretely, they are shown to be equal to the spectral sup-and inf-QFI rates, respectively.This result corresponds to the general formula in entanglement theory [34,35], asserting that the entanglement cost and the distillable entanglement are equal to the spectral sup-and inf-entropy rates.Second, these spectral QFI rates are constructed as asymptotic rates of the smooth max-and min-QFIs.Their construction is parallel to that of spectral entropy rates, given as the asymptotic rates of the smooth max-and min-entropies with the smoothing technique [45,46].Third, the asymmetric majorization relation between energy distribution is shown to provide a necessary and sufficient condition for the exact convertibility among pure states in RTA.This result is the counterpart in RTA to Nielsen's theorem [47], which characterizes the pure-state convertibility in entanglement theory by the majorization relation of the Schmidt coefficients.
Our findings highlight a clear correspondence in noni.i.d.resource conversion in entanglement theory and RTA.See Figs 2 and 3.Although they treat quite different resources, i.e., entanglement and energetic coherence, both are understood within a unified framework of the information-spectrum method for each resource.
Resource theory of asymmetry (RTA).-ThisLetter aims to construct a general theory of manipulating energetic coherence.To this end, we begin by identifying states with and without energetic coherence.Consider a quantum system S and its Hamiltonian H. Energetic coherence means superposition between eigenstates of H with different eigenvalues.Thus, a state ρ has energetic coherence iff the time evolution e −iHt changes it.Conversely, a state without energetic coherence is symmetric under time evolution, i.e., e −iHt ρe iHt = ρ for any t ∈ R. From these facts, we call a state without energetic coherence symmetric and a state with energetic coherence asymmetric.By definition, a state ρ is symmetric iff [ρ, H] = 0.
We next consider transformations of states to manipulate energetic coherence.A basic element is an operation which does not create energetic coherence in the sense that it transforms a symmetric state to a symmetric state.This condition is satisfied if the operation is described by a CPTP map E satisfying [48] E e −iHt ρe iHt = e −iHt E(ρ)e iHt , ∀ρ, ∀t ∈ R. (1) A channel E satisfying Eq. ( 1) is called covariant (under time evolution e −iHt ).
Based on these ideas, RTA is constructed as a resource theory of energetic coherence.The framework of a resource theory is determined by defining "free states" that can be freely prepared and "free operations" that can be freely performed.In RTA, symmetric states are free states, and covariant operations are free operations.With these definitions, energetic coherence in asymmetric states becomes a resource.This structure in RTA is the same as in entanglement theory, where entanglement becomes a resource by defining separable states and local operations and classical communication (LOCC) as free states and free operations.
By adopting the above resource-theoretic perspective, coherence is quantified by resource measures, which monotonically decrease under covariant operations.A well-known and important one is the symmetric logarithmic derivative Fisher information [49,50] with respect to {e −iHt ρe iHt } t , given by where ρ = i λ i |i⟩ ⟨i| is the eigenvalue decomposition.See, e.g., [32,51,52] for details and its generalization.
Hereafter, we call this quantity quantum Fisher information (QFI), simply written as F(ρ).For a pure state, QFI equals four times the variance of H [53].
Following the standard argument [27], we hereafter analyze a system with a Hamiltonian where {|n⟩} denotes an orthogonal basis.With the method in Ref. [27], pure-state conversion theory in this system can be extended to a more general setup in RTA with arbitrary Hamiltonians [54].
An essential characteristic of a pure state ψ = |ψ⟩ ⟨ψ| in the manipulation of energetic coherence is its energy distribution p ψ = {p ψ (n)} ∞ n=0 , where p ψ (n) := | ⟨n|ψ⟩ | 2 .This is because any pure state |ψ⟩ can be mapped to ∞ n=0 p ψ (n) |n⟩ by an energy-conserving unitary operation, which is covariant and invertible.In fact, necessary and sufficient conditions for the exact convertibility between pure states have been obtained in terms of the energy distributions [25,30,55].
From a practical viewpoint, the exact conversion is typically impossible and too restrictive.Therefore, it is common to explore the convertibility with vanishing error in the asymptotic regime.We adopt the trace distance D(ρ, σ) := 1 2 ∥ρ − σ∥ 1 as a quantifier of error, where For ϵ ∈ (0, 1], we denote ρ ≈ ϵ σ iff two states ρ and σ satisfy D(ρ, σ) ≤ ϵ.For two sequences of states ρ = {ρ m } m and σ = {σ m } m , we denote ρ cov ≻ ϵ σ iff there exists a sequence of covariant channels {E m } m such that E m (ρ m ) ≈ ϵ σ m for all sufficiently large m.If ρ cov ≻ ϵ σ holds for all ϵ ∈ (0, 1], we say ρ is asymptotically convertible to σ and denote ρ cov ≻ σ.For simplicity, we only analyze systems with Hamiltonian given by Eq. ( 3).Our main theorem on the pure-state conversion (Theorem 1) for this setup can be extended to a more general setup with arbitrary Hamiltonians.Of course, this includes the i.i.d.case, where a Hamiltonian is given by a sum of copies of a free Hamiltonian of a subsystem.See [54] for a general formula.
In the analysis of asymptotic convertibility, we adopt a coherence bit, i.e., a qubit with Hamiltonian |1⟩ ⟨1| in a state |ϕ coh ⟩ := (|0⟩ + |1⟩)/ √ 2 as a reference.There are two fundamental resource measures: the coherence cost and the distillable coherence.They are defined as the optimal rates for converting a sequence of states from and to coherence bits, i.e., where } m for R > 0 and ϕ coh := |ϕ coh ⟩ ⟨ϕ coh |.Note that the infimum of the empty set is formally defined as +∞.
Finally, we introduce several notations for later convenience.For a = {a(n)} n∈Z , we denote a ≥ 0 iff a For a given sequence q = {q(n)} n∈Z , another sequence q = { q(n)} n∈Z satisfying δ 0,n = q * q(n) (6) plays a central role in our analysis.Here, δ m,n is the Kronecker delta.If there exists a finite n ⋆ := min{n | q(n) > 0}, such a sequence is constructed as Note that q(n) is defined recursively for n > −n ⋆ .If q is an energy distribution, n ⋆ ≥ 0 exists.In this case, q satisfies n q(n) = 1.However, it is not a probability distribution in general since it can contain negative elements.Such a sequence q is utilized to define central quantifiers of our analysis, the max-and min-QFI, just below.
Main results.-Now,let us construct an information-spectrum theory for QFI and show our main results.We first introduce key quantities.For a pure state ψ, we define the max-QFI F max and the min-QFI F min by They quantify the amounts of coherence in ψ transformable from and to a state whose energy distribution follows a Poisson distribution [54].For a general state ρ, we define the max-QFI by F max (ρ) := inf Φρ F max (Φ ρ ), where the infimum is taken over the set of all purifications Φ ρ of ρ and the Hamiltonians of the auxiliary system with integer eigenvalues.This notation is consistent with that for pure states [54].
The max-and min-QFI have similar properties to the max-and min-entropies in entanglement theory [54].For example, they provide the upper and lower bounds for QFI: For a general sequence of pure states ψ = {ψ m }, the spectral sup-and inf-QFI rates are defined as where F ϵ max (ψ) := inf ρ∈B ϵ (ψ) F max (ρ) and F ϵ min (ψ) := sup ϕ∈B ϵ pure (ψ) F min (ϕ) are smooth max-and min-QFI.The main theorem of this Letter is the following: Theorem 1.For a general sequence of pure states ψ = {ψ m }, the coherence cost and the distillable coherence are equal to the spectral sup-and inf-QFI rates, respectively.That is, As a corollary of Theorem 1, we immediately get [54] Replacing F, F and cov ≻ by S, S and LOCC ≻ , the same relations as Eqs.( 14) and (15) hold in entanglement theory.Here, S and S denote the spectral sup-and inf-entropy rates, while ψ LOCC ≻ ϕ means that ψ is asymptotically convertible to ϕ by LOCC [54].
Theorem 1 for a system with Hamiltonian in Eq. ( 3) can be extended to an arbitrary sequence of systems with any Hamiltonians in pure states having a finite period [54].In particular, the spectral QFI rates F and F are equal to QFI F in the i.i.d.setting [54], which reproduces the result in earlier i.i.d.studies [25,27].We remark that S and S are equal to entanglement entropy in the i.i.d.regime in entanglement theory [54].
These results show that the spectral sup-and inf-QFI rates, F and F, in RTA play the same roles as the spectral sup-and inf-entropy rates, S and S, in entanglement theory [54].See Fig. 2. In other words, RTA in the noni.i.d.regime has the same structure on convertibility as Lieb-Yngvason's non-equilibrium theory [56], based on QFI-related quantities rather than entropies.
One-shot convertibility between pure states.-Wehere define a notion of asymmetric majorization, which we abbreviate a-majorization, as follows: Definition 3.For probability distributions p = {p(n)} n∈Z and q = {q(n)} n∈Z , we say that p a-majorizes q iff p * q ≥ 0 hold.In this case, we denote p ≻ a q.
For comparison, we review the definition of majorization.A probability distribution p = {p(i)} d i=1 majorizes another probability distribution q = {q(i) where ↓ indicates that the distributions are rearranged in decreasing order so that p ↓ (i) ≥ p ↓ (j) and q ↓ (i) ≥ q ↓ (j) for i > j.
The a-majorization has properties similar to the ordinary majorization [54].Among them, a significant one is the following: Theorem 4. A pure state ψ is convertible to a pure state ϕ by a covariant operation iff p ψ ≻ a p ϕ .This is the counterpart in RTA to Nielsen's theorem in entanglement theory [47]: A bipartite pure state ψ is convertible to a bipartite pure state ϕ by LOCC iff λ ψ ≺ λ ϕ , where λ ψ and λ ϕ are the probability distributions given by the Schmidt coefficients of ψ and ϕ, respectively.This correspondence is the motivation for introducing the terminology of a-majorization.See Fig. 3.
We remark that other necessary and sufficient conditions on one-shot convertibility in RTA were proven in earlier studies [25,30,55].Our contribution here is to provide the one-shot convertibility condition in terms of a-majorization to make it useful for our purpose to analyze the asymptotic convertibility in the noni.i.d.regime.In particular, this reformulation makes the correspondence between RTA and entanglement theory clearer.
Proof of Theorem 1.-For a Poisson distribution P λ , we denote [54].The followings are key lemmas [54]: Lemma 5. Let E be a covariant channel.A state E(χ λ ) has a purification Ψ such that p Ψ = P λ , where the Hamiltonian of the ancilla added to purify E(χ λ ) has integer eigenvalues.Lemma 6.Let ψ and ϕ be pure states.Assume that a covariant channel satisfies E(ψ) ≈ ϵ ϕ.Then there exists a pure state ψ ′ such that ψ ′ ∈ B 2ϵ 1/4 pure (ψ) and p ψ ′ ≻ a p ϕ .
To show , where we have used the fact that P λ ≻ a P λ ′ holds for any λ ≥ λ ′ [54].From Lemma 5, for all sufficiently large m, there exists a state As ϵ → +0, we get C cost ( ψ) ≥ F( ψ).
To show the opposite inequality, we define 4λ As ϵ → +0, we get Lemma 6, for all sufficiently large m, there exist pure states As ϵ → +0, we get C dist ( ψ) ≤ F( ψ).
To show the opposite inequality, we define 4λ ϵ/2 m := F min (ψ m ).For any δ > 0, there exists δ ′ m , satisfying δ > δ ′ m ≥ 0, such that there exists a pure state As ϵ → +0, we get Conclusion and Discussions.-Inthis Letter, we established the pure-state conversion theory in RTA in the asymptotic non-i.i.d.regime.Unlike entanglement theory, the traditional information-spectrum method for entropy cannot be applied to RTA since its standard measure, QFI, is quite different from entropy.To overcome this issue, we constructed an information-spectrum approach for QFI by carefully analyzing the correspondence between RTA and entanglement theory.It opens the possibility of exploring a unified understanding of asymptotic conversion theory in each branch of quantum resource theories by extending the information-spectrum method for its resource measure.Such an extension may trigger research that has been out of the scope of the information-spectrum method.We speculate that the information-spectrum approach for QFI can be helpful in research areas where QFI plays an essential role, such as in non-equilibrium thermodynamics [57] and general resource theories [58].
The authors thank Achim Kempf for a fruitful discussion.KY acknowledges support from the JSPS Overseas Research Fellowships.HT acknowledges supports from JSPS Grants-in-Aid for Scientific Research (JP19K14610), JST PRESTO (JPMJPR2014), and JST MOONSHOT (JPMJMS2061).Supplemental Material for "Beyond i.i.d. in the Resource Theory of Asymmetry: An Information-Spectrum Approach for Quantum Fisher Information" Koji Yamaguchi 1 and Hiroyasu Tajima The properties of a-majorization ≻ a 3 Properties of F max and F min for pure states 5 F max for general states 7 Proof of Eqs. ( 14) and ( 15) 7 Facts on entanglement theory and spectral entropy rates 8 Interconversion between ϕ coh (R) and χ λ 9 Proof of Lemma 5 11

Poof of Lemma 6 11
Proof of Theorem 2 13 Convertibility between pure states with finite periods 13 General formula for the coherence cost and the distillable coherence for pure states of arbitrary systems 17 In this Supplemental Material, we first complete the proof of theorems in the main text.We then relate the asymptotic convertibility of pure states in harmonic oscillator systems to that of pure states having a finite period.By using these results, we extend Theorem 1 to an arbitrary sequence of systems with any Hamiltonian in pure states having a finite period.Finally, we show that the information-spectral QFI rates are equal to the Fisher information for i.i.d.sequence of pure states with a finite period, which is consistent with the results in the i.i.d.setting [25,27].

The generating function of the Poisson distribution and its reciprocal
For a sequence q = {q(n)} n∈Z , we have defined q as a sequence that satisfies Eq. (6).Although it can be constructed recursively by Eq. ( 7), the method of the generating function [61] is sometimes useful.For simplicity, we assume that n ⋆ = 0.That is, q(n) = 0 for n < 0 and q(0) ̸ = 0.A generating function of q = {q(n)} ∞ n=0 is defined as a formal series given by (S.1) for all z exists if and only if q(0) ̸ = 0 [61].Let a = {a(n)} ∞ n=0 be a sequence generated by 1/f (z).From Eq. (S.2), it satisfies where we have defined a(n) = 0 for n < 0 and δ 0 as a sequence defined by δ 0 = {δ n,0 } n∈Z .In other words, a is the same as the sequence q defined in Eq. (7).We remark that a similar technique, based on characteristic functions instead of generating functions, has been used in [30,55] to derive a necessary and sufficient condition on the one-shot convertibility in RTA.
Let us apply this method to the Poisson distribution.For a sequence {P λ (n)} ∞ n=0 , its generating function is calculated as (S.4) Therefore, its reciprocal is given by the sequence generated by 1/f (z) is given by That is, P λ = P −λ .This result can also be checked directly: Let λ and λ ′ be real parameters.A straightforward calculation shows that (S.8) Since P 0 = δ 0 , we get P λ = P −λ .
Another immediate consequence of Eq. (S.8) is the fact that This is because P σ (n) is non-negative for all n if and only if σ ≥ 0.

Proof of Theorem 4
To prove Theorem 4, a key theorem is the following: , conditions (i) and (ii) are equivalent: (i) There exists a probability distribution {w(k)} k∈Z such that p = k∈Z w(k)Υ k q, where Υ k is a shift operator on probability distribution such that This theorem corresponds to the Hardy-Littlewood-Pólya theorem [62] in the theory of majorization, which states that the following conditions (a) and (b) are equivalent: (a) There exists a doubly stochastic matrix D such that p = Dq.(b) p majorizes q, i.e., p ≻ q.The correspondence becomes more clear by using Birkhoff's theorem [63], which states that any doubly stochastic matrix D can be written as D = k r(k)P k , where {r(k)} is a probability distribution, P k are the permutation matrices, and the sum is taken over the set of all permutation matrices.For details, see, e.g., [64].
Proof of Theorem 7.For a probability distribution {q(n)} n∈Z , the sequence q defined in Eq.( 7) satisfies Eq. ( 6).Therefore, For a probability distribution p = {p(n)} n∈Z , from Eq. ( 6), we have In addition, holds for some w = {w(k)} k∈Z .Then we have i.e., w = p * q.Thus, the sequence w satisfying Eq. (S.17) is unique and given by w = p * q.Since w = p * q satisfies k∈Z w(k) = 1, there exists a sequence {w(k)} k∈Z that satisfies Eq. (S.17), w(k) ≥ 0 and Theorem 4 is obtained as a corollary of Theorem 7 and a theorem in [25] on the convertibility: Theorem 8 (Theorem 3 in [25]).A pure state ψ is convertible to a pure state ϕ by a covariant operation if and only if there exists a probability distribution on integers {w(k)} k∈Z such that p ψ = k∈Z w(k)Υ k p ϕ .

The properties of a-majorization ≻a
We first show that the binary relation ≻ a is a preorder.For any probability distribution p = {p(n)} ∞ n=0 , it holds p ≻ a p since p * p = δ 0 ≥ 0. For probability distributions p = {p(n)} ∞ n=0 , q = {q(n)} ∞ n=0 and r = {r(n)} ∞ n=0 such that p ≻ a q and q ≻ a r, we have for all k ∈ Z, which implies that p ≻ a r holds.Therefore, the binary relation ≻ a is a preorder.It should be noted that the majorization relation ≻ is also a preorder.We further show the following proposition: Proposition 9.For probability distributions p and q, p ≻ a q and q ≻ a p hold if and only if there exists a shift operator Υ k with an integer k ∈ Z such that p = Υ k q.
This corresponds to the fact that λ ≻ λ ′ and λ ′ ≻ λ hold if and only if there exists a permutation matrix P such that λ = P λ ′ in ordinary majorization theory.
Proof of Proposition 9. Let us first show the claim for probability distributions p and q such that n Assume that p * q ≥ 0 and q * p ≥ 0 hold, that is, for all n ≥ 0.
On the other hand, if p = q, then p * q = q * p = δ 0 ≥ 0. Therefore, we have proved the claim under the assumption that n To generalize this result for general probability distributions p and q, consider shifted distributions (S.30) They satisfy n = 0. Note that Eq.( 6) implies (S.31) Since conditions p * q ≥ 0 and q * p ≥ 0 are equivalent to p ′ * q ′ ≥ 0 and q ′ * p ′ ≥ 0, they hold if and only if or equivalently, Properties of Fmax and Fmin for pure states Let us prove the monotonicity of the variance under ≻ a .Let p and q be probability distributions such that p ≻ a q.Since p = k∈Z w(k)Υ k q for a probability distribution w = p * q ≥ 0, we have = µ w + µ q , (S.36) Similarly, it holds that ≥ Var(q), (S. 40) where in the last line, we have used w(k) ≥ 0 for all k.Of course, this monotonicity is expected from the fact that QFI monotonically decreases under a covariant operation and that QFI is four times the variance for pure states.Equation ( 10) is a consequence of this monotonicity.To prove it, let us first show an easy but useful lemma: Lemma 10.Let ψ be a pure state.If F max (ψ) < +∞, then for any λ such that 4λ > F max (ψ), it holds P λ ≻ a p ψ .Similarly, if F min (ψ) > 0, then for any σ such that F min (ψ) > 4σ ≥ 0, it holds p ψ ≻ a P σ .
Furthermore, it should be noted that when the energy distribution of a pure state ψ is given by a Poisson distribution, it holds (S.42) We also prove the following proposition, which corresponds to Eqs. ( 14) and ( 15) in the one-shot regime: Proposition 12.For any pure states ψ and ϕ, we have ( Let λ be any real number such that 4λ > F max (ψ).From Lemma 10, this means P λ ≻ a p ψ .Since ≻ a is a preorder, it implies that P λ ≻ a p ϕ and therefore 4λ ≥ F max (ϕ).Since λ is an arbitrary real number such that 4λ > F max (ψ), we have F max (ψ) ≥ F max (ϕ).
In entanglement theory, a maximally entangled state is a reference state adopted in the literature, where d denotes the dimension of the Hilbert spaces for each subsystem A and B, while {|i⟩ Z } d i=1 is an orthonormal basis for each subsystem Z = A, B. The reduced state is the maximally mixed state where I d denotes the identity operator.In this case, we have This corresponds to Eq. (S.42).
If a state ρ is convertible to another state σ by LOCC, we denote ρ LOCC ≻ σ.An important theorem on the one-shot convertibility between pure states by LOCC is the following: Theorem 13.Let ψ AB and ϕ AB be bipartite pure states.Define the reduced states ρ A := Tr B (ψ AB ) and σ A := Tr(ϕ AB ).The following two hold: This is the counterpart of Proposition 12.

Fmax for general states
In the main text, we have defined where the infimum is taken over the sets of all purifications Ψ ρ of ρ and Hamiltonians H A with integer eigenvalues of the auxiliary system A that is added to purify ρ.
Proof of Eqs. ( 14) and (15) Equations ( 14) and ( 15) are obtained as a corollary of Theorem 1 and the following proposition: Proposition 14.For any sequences of pure states ψ = {ψ m } m and ϕ = {ϕ m } m , the followings hold: (1) Proof.(1): For R := C cost ( ψ) and any positive number δ > 0, there exists δ ′ such that δ > δ ′ ≥ 0 and (S.50) Since ψ cov ≻ ϕ holds from the assumption, we have Since δ > 0 is arbitrary and δ > δ ′ , we have C cost ( ψ) ≥ C cost ( ϕ).In a similar way, C dist ( ψ) ≥ C dist ( ϕ) is proven. ( Similarly, there exists a real number R ′′ such that R > R ′′ ≥ C cost ( ϕ) and Facts on entanglement theory and spectral entropy rates We here provide results in entanglement theory in the literature without proof.Let ψ AB and ϕ AB be bipartite pure states.We define the density operators for the subsystem A as In this case, we denote ψ AB LOCC ≻ ϕ AB (R).It is known [33] that the conversion from In other words, pure states are interconvertible and the optimal rate is given by the ratio of the entanglement entropies in the i.i.d.regime.
To analyze the asymptotic convertibility in a more general setup, it is common to adopt a maximally entangled state as a reference.Let us first define the entanglement cost.For a given sequence of pure states ψ = {ψ m } m , we say that a rate R is achievable in a dilution process if and only if there exists a sequence of nonnegative numbers where we have defined Φ({N m }) := {Φ Nm } m for the maximally entangled state defined in Eq. (S.46).The entanglement cost of ψ is defined by In a similar way, we can define the distillable entanglement.We say that a rate R is achievable in a distillation process if and only if there exists a sequence of nonnegative numbers The distillable entanglement is defined as The spectral sup-and inf-entropy rates in the quantum case have been developed in different contexts, e.g., in [34,35,39,40,45,65].Here we provide one of the alternative but equivalent definitions, which is based on the smoothing technique [45,46].For a given sequence of states ρ = {ρ m } m , its spectral sup-and inf-entropy rates are defined by For a sequence of general pure states ψ AB = {ψ AB,m } m , let us define a sequence of reduced states by ρ A = {ρ A,m } m , where ρ A,m := Tr B (ψ AB,m ).It is shown [34,35] that

S( ρ) := lim
(S.62) For the convertibility between sequences of pure states ψ AB = {ψ AB,m } m and ϕ AB = {ϕ AB,m } m , the following two hold [44]: where we have defined σ A := {σ A,m } m and σ A,m := Tr B (ϕ AB,m ).They are the counterparts of Eqs. ( 14) and ( 15) in entanglement theory.In particular, for an i.i.d.sequence of pure states ψ AB = {ψ ⊗m AB } m , the spectral entropy rates are equal to the entanglement entropy: (S.65) Interconversion between ϕ coh (R) and χ λ We here show the following: Lemma 15 is proved based on the arguments in [27].The following lemma connects the closeness of energy distributions and the convertibility of pure states.Lemma 16 ([27]).Let ψ and ϕ be pure states.There exists a covariant unitary operator U such that (S.66) Here, the total variation distance between two probability distributions is defined by In the i.i.d.regime, the translated Poisson distribution plays an important role, which is defined as follows: Definition 17 (The translated Poisson distribution).For µ ∈ R and σ 2 ≥ 0, the translated Poisson distribution is defined by where s := ⌊µ − σ 2 ⌋ is an integer and γ := µ − σ 2 − ⌊µ − σ 2 ⌋ is a parameter satisfying 0 ≤ γ < 1.The mean and variance are given by µ and σ 2 + γ, respectively.Alternatively, the translated Poisson distribution is written as It is known that the sum of integer-valued random variables converges to the translated Poisson distribution [27,66].Let {Z i } m i=1 be a set of independent random variables with mean µ i := EZ i and variance σ 2 i := E((Z i − µ i ) 2 ).Assume that its absolute third moment E|Z 3 i | is finite.Let L(Z) denote the probability distribution of a random variable Z.We define ) , (S.69) For the sum of the random variables W := m i=1 X i with mean μ := E(W ) = m i=1 µ i and variance σ2 : i , the following theorem holds: Theorem 18 (Corollary 3.2 in [66], Theorem 7 in [27]).Suppose that the random variable For Poisson distributions with different variance, the upper bound on the total variation distance is provided in the following theorem: Theorem 19 (Equation (5) in Ref. [67], Equation (2.2) in Ref. [68], Lemma 8 in Ref. [27]).
Rm ≻ a q m and q m ≻ a P 1 4 Rm hold, there exist covariant channels Λ m and Λ ′ m such that χ 1 4 Rm = Λ m (κ m ) and κ m = Λ ′ m (χ 1 4 Rm ).On the other hand, according to Lemma 16, there exist covariant channels Θ m and Θ ′ m such that Proof of Lemma 5 Proof.Let E be a covariant channel with respect to the Hamiltonian given by Eq. ( 3).From the covariant Stinespring dilation theorem [30,69], there exists an ancillary system A, a symmetric pure state η A of A, the Hamiltonian H A and a covariant unitary operator U such that In our setup, the Hamiltonian of the original system is assumed to be given by Eq. ( 3).In this case, the Hamiltonian H A of the auxiliary system has integer eigenvalues.Furthermore, without loss of generality, it is possible to assume that η A is an energy eigenstate with a vanishing eigenvalue of H A , i.e., H A |η A ⟩ = 0. For Since U is covariant and H A |η A ⟩ = 0 holds, we have p Φ = p χ λ = P λ .In addition, Φ is a purification of E(χ λ ), which concludes the proof of Lemma 5.
Lemma 5 shows that F ∞ (ρ) for a general state ρ quantifies the minimum amount of coherence in χ λ required to generate ρ.

Poof of Lemma 6
Instead of Lemma 6, we here prove a slightly improved lemma: Lemma 20.Let ψ and ϕ be pure states.If there exists a covariant channel E such that E(ψ) ≈ ϵ ϕ for 0 ≤ ϵ ≤ 1, then there exists a pure state ψ ′ such that In the proof Lemma 20, we use the following: For a given pure state ψ, we define For each (k, u) such that q (k,u) ̸ = 0, we define a normalized pure state The energy distributions for pure states are related as or equivalently, where we have defined For this probability distribution {w(k)} k∈Z , we have (S.99) By using the the Fuchs-van de Graaf inequalities, we have (S.105) Therefore, we finally get Proof of Lemma 20.From Lemma 21, there exists a probability distirubiton {w(k)} k∈Z such that (S.107) Defining q := w * p ϕ and |ψ ′′ ⟩ := n q(n) |n⟩, from Lemma 16, there exists a covariant unitary operator U such that Since q ≻ a p ϕ and p ψ ′ ≻ a q, we have p ψ ′ ≻ a p ϕ .

Proof of Theorem 2
Theorem 2 can be proven in the same way as the first part of Theorem 1.For completeness, we here repeat the proof.

Convertibility between pure states with finite periods
So far, we have constructed asymptotic conversion theory in RTA in the non-i.i.d.regime under the assumption that the Hamiltonians are given by the one for a harmonic oscillator system.Here, improving the arguments in [25,27], we show that the convertibility between pure states with finite periods can be analyzed with a harmonic oscillator system with the Hamiltonian in Eq. (3).
To begin with, we analyze a one-shot setting.Let   In the following, we analyze the conversion from a pure state in A to a pure state in B. For a pure state ψ A , its period is defined by Assuming that τ is finite and non-zero, a state ψ A is mapped by a covariant operation to a state with a finite period τ /k for some positive integer k or a symmetric state.Of particular interest here is the former case since the latter case is trivial as any state can be mapped to any symmetric state with a covariant operation.Therefore, we analyze the convertibility under the assumption that ϕ B has a finite period τ ′ = τ /k with some positive integer k.
When the pure state ψ A has a finite period τ , the eigenvalues in Spec ψ (H A ) are expressed as 2π τ n + E 0 with some n ∈ Z ≥0 and a constant E 0 .Shifting the Hamiltonian by a constant, without loss of generality, we can set E 0 = 0.In the same way, we introduce for a pure state ϕ B .When the pure state ϕ B has period τ /k for some positive integer k, shifting the Hamiltonian H B by a constant, without loss of generality, all eigenvalues in Spec ϕ (H B ) are expressed as 2π τ kn for some n ∈ Z ≥0 .Finally, by multiplying the Hamiltonians H A and H B by τ /(2π), we can assume that Spec ψ (H A ) ⊂ Z ≥0 and Spec ϕ (H B ) ⊂ Z ≥0 .
For pure states ψ A and ϕ B , we define their energy distributions by By using the energy distributions, the pure states ψ A and ϕ B are expanded as where H HO = Span{|n⟩ HO } ∞ n=0 denotes the Hilbert space for the harmonic oscillator system.We now prove a lemma on one-shot convertibility among pure states with and without an error.
Lemma 22.Let ψ A and ϕ B be pure states on A and B with finite periods τ and τ ′ , respectively.Assume that τ ′ = τ /k for some positive integer k.Define pure states ψ ′ HO and ϕ ′ HO in a harmonic oscillator system by Eq. (S.119).Then, the following two hold: Proof.Let us first introduce maps relating the original systems A and B to a harmonic oscillator system.Defining operators |n⟩ HO ⟨ϕ n | B : H B → H HO , (S.120) we have Furthermore, they satisfy We introduce maps HO : L(H B ) → L(H HO ), (S.125)Note that claim (ii) in this theorem is valid even when one adopts any quantum distance satisfying the dataprocessing inequality instead of the trance distance to quantify the error.
We here extend the above arguments to asymptotic convertibility.Let ρ A = {ρ Am } m and σ B = {σ Bm } m be sequence of states of systems A m and B m , respectively.We assume that the Hamiltonians H A = {H Am } m and H A = {H Bm } m are bounded below.For ϵ ∈ (0, 1], we denote ( ψ A , H A ) Note that if we multiply all the Hamiltonians by τ 2π , we can assume that the period is 2π without loss of generality.By using the sequence of states ψ = {ψ m }, we define F ψ, H := F ψ HO , F ψ, H := F ψ HO , (S.149) where the right hand sides are the spectral sup-and inf-QFI rates defined in Eqs.(11) and (12) in the main text.
From Theorem 1 and Corollary 23, we get a general formula for the coherence cost and the distillable coherence for pure states in general systems with arbitrary Hamiltonians: By using the results in [70], Eq. (S.158) can be proven as a consequence of asymptotic monotonicity of smooth metric adjusted skew information rates.
with some parameters a, b and c.Thend TV L(W ), TP μ,σ 2

cov≻
ϵ σ by omitting the Hamiltonian as in the main text.
cov ≻ ϵ ( ϕ B , H B ) if and only if there exists a sequence of covariant operations {E m } m such that D(E m (ρ m ), σ m ) ≤ ϵ for all sufficiently large m.If ( ψ A , H A ) cov ≻ ϵ ( ϕ B , H B ) holds for all ϵ ∈ (0, 1], then we denote ( ψ A , H A ) cov ≻ ( ϕ B , H B ).If no confusion arises, we omit Hamiltonians and simply write ρ cov ≻ ϵ σ and ρ cov ≻ σ as in the main text.As a corollary of Lemma 22, we prove that the asymptotic convertibility among pure states can be analyzed with harmonic oscillator systems.

Theorem 24 . 4 F
For any sequences of pure states ψ = {ψ m } and Hamiltonians H = {H m } with a finite period, it holdsC cost ψ, H = F ψ, H , C dist ψ, H = F ψ, H . (S.150)As a consistency check of the result in the i.i.d.regime[25,27], we show thatF ψ iid , H iid = F ψ iid , H iid = F(ψ, H) (S.151) for i.i.d.sequence of pure states ψ iid = {ψ ⊗m } and Hamiltonians { H iid } = {H iid,m } with period 2π, where H iid,m := m i=1 I ⊗i−1 ⊗ H ⊗ I ⊗m−i .First, following Eq.(S.147), let us introduce |ψ HO,m ⟩ := n∈Spec ψ ⊗m (H iid,m ) p ψ ⊗m (n) |n⟩ HO (S.152) generalizing Eq. (S.77), we find that lim m→∞ d TV p ψ ⊗m , P m 1 (ψ,H) = 0, (S.153) implying that ∀ϵ ∈ (0, 1], mF(ψ, H) ≥ F ϵ max (ψ HO,m ), F ϵ min (ψ HO,m ) ≥ mF(ψ, H) (S.154) hold for all sufficiently large m.Therefore, we get F(ψ, H) ≥ F ψ iid , H iid , F ψ iid , H iid ≥ F(ψ, H). (S.155) Thus, in order to prove Eq. (S.151), it is sufficient to show F ψ, H ≥ F ψ, H , (S.156) or equivalently, C cost ψ, H ≥ C dist ψ, H . (S.157)Although Eq. (S.157) might seem to be trivial, technically, one must prove ϕ coh (R) cov ≻ ϕ coh (R ′ ) =⇒ R ≥ R ′ .(S.158) Here, we defined B ϵ (ψ) := {ρ : states|D(ρ, ψ) ≤ ϵ} and 2,3 1 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 2 Department of Communication Engineering and Informatics, University of Electro-Communications, 1-5-1 Chofugaoka, A and B be quantum systems associated with Hilbert spaces H A and H B .We assume that their Hamiltonians H A and H B are bounded from below.Let L(H) denote the set of all linear operators on a Hilbert space H.A quantum channelE A→B : L(H A ) → L(H B ) is called covariant if and only if e −iH B t (E A→B (ρ A )) e iH B t = E A→B e −iH A t ρ A e iH A t (S.111)holds for all states ρ A of the system A and for all t ∈ R. Note that the covariance is defined with respect to the Hamiltonians of the input and output systems of the channel.Clarifying this point, we denote (ρ A , H A ) B , H B ) if and only if there exists a covariant channel E A→B such that E A→B (ρ A ) = σ B .Similarly, we denote (ρ A , H A ) B , H B ) for ϵ ∈ (0, 1] if and only if there exists a covariant channel E A→B such that D(E A→B (ρ A ), σ B ) ≤ ϵ.When no confusion arises, we simply write ρ cov ≻ (σ cov ≻ σ and ρ A) Ebe the spectral decomposition of the Hamiltonian, where Spec(H A ) is the set of different eigenvalues of H A and Π denotes the projector to the eigenspace with eigenvalue E. For a given pure state ψ A , we define Spec ψ (H A S.117) where |ψ n ⟩ A and |ϕ n ⟩ B , are eigenvectors of H A and H B with eigenvalue n, respectively.Let us introduce a harmonic oscillator system whose Hamiltonian is given by ′ ⟩ HO := n∈Spec ψ (H A ) p ψ (n) |n⟩ HO ∈ H HO , |ϕ ′ ⟩ HO := n∈Spec ϕ (H B ) p ϕ (n) |n⟩ HO ∈ H HO ,(S.119)