Initial-State Dependence of Thermodynamic Dissipation for any Quantum Process

Herein we provide a new exact result about the nonequilibrium thermodynamics of open quantum systems at arbitrary timescales. In particular, we show that the contraction of the quantum state---towards the minimally-dissipative trajectory---exactly quantifies the excess of thermodynamic dissipation during any finite-time transformation. The quantum component of this dissipation is the change in coherence relative to the minimally-dissipative state. Implications for quantum state preparation, local control, and decoherence are explored. For logically-irreversible processes---like the preparation of any particular quantum state---we find that mismatched expectations lead to divergent dissipation as the actual initial state becomes orthogonal to the anticipated one.

Much recent progress extends Landauer's principle to the quantum regime-affirming that quantum information is physical [1][2][3][4][5]. Associated bounds refine our understanding of how much heat needs to be exhausted-or how much work needs to be performed, or could be extractedto preserve the Second Law of Thermodynamics: Entropy production is expected to be non-negative Σ ρ0 ≥ 0 from any initial density matrix ρ 0 . However, these Landauertype bounds only become tight in the infinite-time quasistatic limit, as entropy production goes to zero. Yet infinite time is not a luxury afforded to quantum systems with short decoherence time. And, even if coherence can be maintained for significant time-length, we want to know the thermodynamic limits of both quantum computers and natural quantum processes that transform quickly. Unfortunately, very few exact general results are known about the finite-time nonequilibrium thermodynamics of open quantum systems. The short list includes quantum generalizations of Crooks' Relation [6,7], the Jarzynski Equality [8,9], and other fluctuation relations [10][11][12][13], which are nonequilibrium equalities that in many ways subsume the inequality of the Second Law of Thermodynamics. More often we must rely on approximations, like the linearresponse and local-equilibrium theories developed many decades ago [14]; and like the weak-coupling, Markovian, and other approximations that lead to quantum Markovian master equations [15,16]. Despite the practical successes of these approximations [17][18][19], further exact results are highly desirable since they could yield new fundamental insight into the interplay among quantum correlations, dissipation, and other aspects of physics.
Here we add to the short list of exact general results for quantum finite-time dissipation. We will show that the thermodynamic dissipation due to alternative initial density matrices is exactly quantified for any finite-time transformation by the contraction of the relative entropy between the actual density matrix ρ t and the minimallydissipative density matrix q t . This generalizes a recent theorem by Kolchinsky and Wolpert, demonstrating that quantum coherence plays a significant role in the quantum extension of this previous classical result [20].
We illustrate some immediate consequences. The first being a thermodynamic risk in overspecialization despite the reward for specialization: To minimize heat dissipation, one should tailor the implementation of a desired quantum operation to particular initial state distributions; but the same optimization can lead to risks of divergent dissipation when input states differ significantly from predictions. The second is the thermodynamic costs of modularity: where quantum operations optimized for thermal efficiency locally can result in unavoidable dissipation beyond Landauer's limit when acting on correlated systems. All of these results are valid over arbitrarily short timescales.
Background To proceed, we consider any quantum system with initial density matrix ρ 0 . We allow a set B of canonical or grand canonical baths to play an active role in the evolution of the system over the duration τ . A control protocol x 0:τ determines the time-dependent interaction H int xt with the thermal and chemical baths (mediating heat and particle exchange) and also controls the time-dependent Hamiltonian H xt of the system to perform work on the system during the protocol.
The initial joint state of the system and baths is assumed arXiv:2002.11425v1 [cond-mat.stat-mech] 26 Feb 2020 to be separable: We furthermore assume each bath is initially in canonical or grand canonical equilibrium: ρ are the chemical potential and the number operator for the bath's th particle type, and Z (b) is the grand canonical partition function for the bath [21]. The baths are assumed to be sufficiently large that their temperature and chemical potentials remain unchanged throughout the protocol.
The driving protocol x 0:τ induces a net time evolution U x0:τ of the system-baths mega-system 1 , so that the joint state is given at the end of the transformation as: We will also consider the reduced density matrices of the system ρ τ = tr B (ρ tot τ ) and baths ρ (b) τ = tr sys,B\b (ρ tot τ ) after the joint evolution. Thermodynamic dissipation is quantified by entropy production Σ ρ0 , which is the effectively-irreversible change in entropy 2 . It is the entropy flow to the environment beyond any reduction in the entropy of the system [11,14,22,23]: The expected entropy flow to the environment over the course of the process is [14,[22][23][24]: where the heat Q (b) is the energy change of bath b over the course of the process: ). The expected change in the thermodynamic entropy of the system is proportional to its change in von Neumann 1 While we only utilize the existence of the net unitary time evolution, we note that it is induced through the time-ordered exponential: Some authors use the term 'dissipation' more loosely, to refer to heat even when it is not associated with entropy production. In either usage, the entropy production quantifies the dissipation beyond the minimal requirements of the Second Law. entropy 3 [11,24]: Since the von Neumann entropy of the joint universe is unchanged under unitary dynamics, entropy production can be viewed as the change in total correlation built up among the system and baths [23,24]-correlations that are too difficult to leverage.

Derivation of Main Result
The initial density matrix ρ 0 can be represented in an arbitrary orthonormal basis as: with c j,k = j| ρ 0 |k . We can consider all possible variations of the initial density matrix via changes in these c j,k parameters.
We aim to expose the ρ 0 -dependence of entropy production. From Eqs. (2) and (5), it is easy to show that [24]: τ . Meanwhile, utilizing the spectral theorem, it is useful to rewrite the change in system entropy as: (9) where Λ ρ0 is the collection of ρ 0 's eigenvalues, and Λ ρτ is the collection of ρ τ 's eigenvalues. We then calculate the infinitesimal perturbations ∂λ0 ∂c j,k and ∂λτ ∂c j,k . As shown in Appendix A, varying a single parameter c j,k of the initial density matrix yields the partial derivative: To consider the consequences of arbitrary variations in the initial density matrix, we construct a type of gradient ∇Σ ρ0 ≡ j,k |k j| ∂ ∂c j,k Σ ρ0 with a scalar product "·" that gives a type of directional derivative: γ · ∇Σ ρ0 ≡ tr(γ∇Σ ρ0 ).
For any two density matrices, (ρ 0 − ρ 0 ) · ∇Σ ρ 0 gives the linear approximation of the change in entropy production (from the gradient evaluated at ρ 0 ) if we were to change the initial density matrix from ρ 0 towards ρ 0 . Notably, using our directional derivative this way allows us to stay along the manifold of density matrices (due to the convexity of quantum states), while inspecting the effect of all possible infinitesimal changes to ρ 0 . Lemma 1. For any two density matrices ρ 0 and ρ 0 : Lemma 1 follows directly from Eqs. (7) and (10). Hence, for any initial density matrix: It is worthwhile to consider the density matrix q 0 that would lead to minimal entropy production under the control protocol x 0:τ . By definition of q 0 ∈ argmin ρ0 Σ ρ0 (11) as an extremum: if (on the one hand) q 0 has full rank, it must be true that: for any density matrix ρ 0 . I.e., moving from q 0 infinitesimally in the direction of any other initial density matrix cannot produce a linear change in the dissipation. Expanding Eq. (12), ρ 0 · ∇Σ q0 − q 0 · ∇Σ q0 = 0, according to Lemma 1 yields our main result: where D[ρ q] ≡ tr(ρ ln ρ)−tr(ρ ln q) is the relative entropy.
If (on the other hand) argmin ρ0 Σ ρ0 has a nontrivial nullspace, then Eq. (13) can be extended by supplementing q 0 with the successive minimally dissipative density matrices on the nullspace. This extension of our main result is derived and discussed in Appendix C.
We see that the information loss D ρ 0 q 0 −D ρ τ q τ generalizes and quantifies the notion of logical irreversibility relevant to physics.

Theorem 1.
Any logically irreversible operation requires dissipation-of at least k B D ρ 0 q 0 −k B D ρ τ q τbeyond Landauer's bound, and cannot simultaneously be thermodynamically optimized for all initial states.
It is interesting to compare to the classical result by Kolchinsky and Wolpert [20], which gives this dissipation in terms of the Kullback-Leibler divergence D KL instead of the quantum relative entropy D. The correspondence is complicated by the fact that q 0 and q τ are not typically diagonalized in the same basis. Nevertheless, we can consider q t 's eigenbasis at each time, and describe ρ t 's coherence in this basis as well as the classical probability distribution that would be induced by projecting ρ t onto q t 's eigenbasis. In particular, the classical probability distribution that would be induced by projecting ρ t onto q t 's eigenbasis is: where each |s is an eigenstate of q t . Since P t and q t are diagonal in the same basis, the relative entropy D P t q t reduces to the Kullback-Leibler divergence D KL P t q t . However the actual density matrix ρ t is typically not diagonalized by q t 's eigenbasis-but rather exhibits coherence there. This coherence is naturally quantified by the so-called 'relative entropy of coherence' [25]: when we take the incoherent basis to be the eigenbasis of the minimally-dissipative density matrix. As shown in Appendix D, the extra dissipation from starting with the density matrix ρ 0 rather than the minimallydissipative q 0 is given for any finite-duration nonequilibrium transformation as: We see that the quantum correction to the classical dissipation is exactly the change of coherence on the minimallydissipative eigenbasis. We now consider several important cases before turning to further general results. Relaxation to equilibrium Let us first consider the simple case of constant weak coupling to a single thermal bath of inverse temperature β = 1 k B T . Suppose the system is undriven, so that it experiences a time-independent Hamiltonian H x . There is zero dissipation if the system starts in equilibrium, so q 0 = π x = e −βHx /Z (where Z is the system's canonical partition function), with Σ πx = 0. The dissipation when starting in state ρ 0 is thus: which is a well known result since F add t = k B T D ρ t π xt is the system's nonequilibrium addition to free energy [11,[26][27][28]. The unharnessed reduction in the nonequilibrium addition to free energy results in entropy production. Furthermore, we see that this dissipation can be decomposed as: , into contributions from the change in probability of the system's energy eigenstates D KL P 0 π x − D KL P τ π x and from the decoherence in the energy eigenbasis: Reset Consider any control protocol x 0:τ that implements RESET to a desired state r τ from all initial quantum states ρ 0 . 4 For example: to erase any number of qubits (or qutrits, etc.) or, similarly, to initialize an entangled Bell state. It is reasonable to desire that the control protocol x 0:τ achieves this erasure with fidelity F (ρ τ , r τ ) ≥ 1 − for all possible ρ 0 , for some allowable error tolerance 0 ≤ 1.
Since D ρ τ q τ = 0 + O( ) for the RESET operation, our Eq. (13) asserts that implementing erasure with the fixed protocol x 0:τ must result in dissipation for any ρ 0 = q 0 . In particular, with fidelity better than 1 − , then: It should be emphasized that this dissipation is distinct and additive to the Landauer cost of erasure [3,4], the latter of which is achieved in the limit of zero dissipation.
For thermodynamic efficiency, the reset protocol must be designed around the expected initial state. But what if the initial state is unknown, or expectations are misaligned? Fig. 1 illustrates the thermodynamic cost of misaligned expectations when the RESET operation is applied to two qubits that the protocol is not optimized for. This exposes the risk of divergent dissipation upon overspecializationwhen the protocol operates on a state that is nearly orthogonal to the anticipated initial state q 0 .
For energetic efficiency in multiple use cases-say in resetting unknown qubits of a quantum computer-it is advisable when constructing such protocols to hedge thermodynamic bets. Bringing q 0 closer to the identity assures that no state is orthogonal to it.
Local control of composite systems Often, only local transformations are applied to globally correlated systems. For example, when computing, it is convenient to apply modular logic gates to implement complicated net transformations. Our main result implies thermodynamic consequences of this local control.
Suppose our system of interest is composed of N subsystems that live on the composite Hilbert space H = Whether preparing a Bell state, erasing quantum memory, or extracting work, no control protocol can be simultaneously thermodynamically optimal for all initial quantum states on which it operates. Inset: A quantum system ρt of two qubits is driven by a time-dependent Hamiltonian Hx t and a time-dependent interaction H int x t with two thermal reservoirs at different temperatures. Main: Suppose the control protocol x0:τ resets the two qubits in finite time, and achieves minimal dissipation when operating on the noisy Bell state: If the reset protocol is designed to be optimal for erasing the Ψ + Bell state (α = 0), then the same protocol approaches infinite dissipation as ρ0 approaches any of the Φ + , Φ − , or Ψ − Bell states orthogonal to it. For 0 < α 1, the dissipation Σρ 0 scales as ln(1/α) near these three Bell states. Whereas if α = 1-i.e., the reset protocol is optimized for erasing completely randomized classical bits-then any quantum state can be erased with no more than 2 ln(2)k B of dissipation beyond Σq 0 (where Σq 0 can be engineered to be arbitrarily small). This dissipation is distinct and additive to the Landauer cost of erasure.
N n=1 H n . The initial state ρ 0 may have both classical and quantum correlations among its constituent parts. Define ρ 0,n = tr H\Hn (ρ 0 ) to be the reduced density matrix of the n th subsystem. For any control protocol that is locally optimized, such that q 0 = N n=1 ρ 0,n and q τ = N n=1 ρ τ,n , the dissipation is the loss in total correlation: In the case of N = 2 subsystems, this is the change in quantum mutual information between them: ∆I[ρ t,1 ; ρ t,2 ].
This will be relevant when only local control is applied to globally correlated systems, when the baths do not further correlate the subsystems [29,30]. This implies dissipation from destroyed correlations during both local measurement and local erasure of entangled systems. And, since computations are often performed modularly, this may be an important contribution to heat generated during a logically irreversible computation. Further dissipation is expected when the system is not locally optimized.
Decoherence The process of decoherence implements the map ρ 0 → m Π m ρ 0 Π m . With minimal physical assumptions, it is plausible that a minimally dissipative initial state would be q 0 = m Π m ρ 0 Π m , since decoherence leaves this state unchanged. In such cases: Discussion We have produced a useful general result that exactly quantifies dissipation in finite-duration transformations of open quantum systems. When the system is initiated in any state other than the minimally-dissipative density matrix, the extra dissipation is exactly the contraction of the quantum relative entropy between them over the duration of the control protocol-the loss of distinguishability.
This has immediate consequences for thermally efficient quantum information processing. Crucially, a quantum control protocol cannot generally be made thermodynamically optimal for all possible input states, creating unavoidable dissipation beyond Landauer's in quantum state preparation. Meanwhile, it imposes extra thermodynamic cost to modular computing architectures, where one wishes to optimize the thermal efficiency of certain quantum operations without pre-knowledge of how they will fit within a composite quantum protocol.
Our results also complement related but distinct results on the initial-state dependence of free energy gain [31]. Appendix I unifies these results, providing a more general theory of state-dependence in energetic computation. Since our results accommodate arbitrary interactions with any number of thermochemical baths, they could be leveraged in future studies to analyze dissipation in relaxation to nonequilibrium steady states (see Appendix G). From a broader perspective, these results extend our understanding of effective irreversibility in quantum mechanics, despite its global unitarity.
Varying a single parameter c j,k of the initial density matrix yields the partial derivative: which leads us to evaluate the infinitesimal perturbations ∂λ0 ∂c j,k and ∂λτ ∂c j,k to the eigenvalues of ρ 0 and ρ τ respectively.
(We could alternatively choose to vary real-valued variables c (r) j,k and c j,k ) |j k| and differentiate with respect to these real variables. However, it conveniently turns out that Σ ρ0 is complex-differentiable in all complex-valued c j,k variables, so we can differentiate directly with respect to c j,k .) Starting with the eigen-relation: ρ 0 |λ 0 = λ 0 |λ 0 , we can take the partial derivative of each side: Left-multiplying by λ 0 |, and recalling that ρ 0 = j,k c j,k |j k|, we obtain: which yields The summations over λ 0 thus become: and Moving on to the slightly more involved perturbation, we use the eigen-relation: ρ τ |λ τ = λ τ |λ τ and again take the partial derivative of each side: Left-multiplying by λ τ |, and recalling that ρ τ = j,k c j,k tr B U x0:τ |j k| ⊗ b π (b) U † x0:τ , we obtain: which yields The summations over λ τ thus become: and λτ ∈Λρ τ Plugging in our new expressions for the λ 0 an λ τ summations in Eqs. (26), (27), (32), and (35), we obtain: so that we arrive at Eq. (10) of the main text.
This allows us to inspect local changes in entropy production (ρ 0 − ρ 0 ) · ∇Σ ρ 0 as we move from ρ 0 towards any other density matrix ρ 0 . By the convexity of quantum states, there is indeed a continuum of density matrices in this direction; so the sign of the directional derivative indeed indicates the sign of the change in entropy production for infinitesimal changes to the initial density matrix in the direction of ρ 0 .

Lemma 2. Dissipation is convex over initial density matrices.
Let ρ 0 = n Pr(n)ρ Eq. (37) is obtained from the recognition that entropy flow ∆S env is an affine function of ρ 0 , and so cancels between Σ ρ0 and n Pr(n)Σ ρ [n] 0 . Each of the square brackets then represents a Holevo information, before and after the transformation, respectively. The non-negativity is finally obtained by the information processing inequality applied to the Holevo information. (Non-negativity can be seen to follow from each n separately: I.e., D ρ with multiplicities inherited from the constituent spectra.

APPENDIX C: GENERALIZED q0 FOR NON-INTERACTING BASINS
It is possible that the evolution acts completely independently on distinct basins of state space. This will generically yield a nontrivial nullspace for argmin ρ0 Σ ρ0 . In such cases, it is profitable to generalize the definition of q 0 so that it includes the successive minimally-dissipative density matrices that carve out the independent basins on the nullspace.
We will show that, within each basin, the extra dissipation due to a non-minimally-dissipative initial density matrix is given exactly by the contraction of the relative entropy between the actual and minimally-dissipative initial density matrices under the same driving x 0:τ .
To make progress in this generalized setting, we must first introduce several new notions.

Definitions
Let P(H) be the set of density matrices that can be constructed on the Hilbert space H. I.e., We will denote the nullspace of an operator ρ as null(ρ). I.e.: null(ρ) = |η : ρ |η = 0 . Furthermore, let H sys be the Hilbert space of the physical system under study (not including the environment).
We can now introduce the successive minimally dissipative density matrices {q In the main text, where q has a nontrivial nullspace, we will also want to consider the minimally dissipative density matrix on the nullspace: q [1] 0 ∈ argmin ρ0∈P null(q Σ ρ0 . If q [1] 0 also has a nontrivial nullspace, then we continue in the same fashion to identify the minimally dissipative density matrix within the intersection of all of the preceding nullspaces. In general, the n th thermodynamically-independent basin has the minimally dissipative initial state: for n ≥ 1.
The n th minimally dissipative basin is the Hilbert space: H 0 \ 0 . We will employ the projector: 0 . Notably, these projectors constitute a decomposition of the identity I on the system's state space H sys : We can now define the minimally dissipative reference state q 0 , as if ρ 0 were minimally dissipative on each of the thermodynamically-independent basins on which it lives: It should be noted that the ρ 0 -dependence is only via the weight tr(Π H [n] 0 ρ 0 ) of ρ 0 on each thermodynamicallyindependent basin, used to linearly combine their contributions.

Generalized dissipation bound
With these definitions in place, let us now reconsider the task at hand.
If q [0] 0 has a nontrivial nullspace (and if Σ ρ0 is finite for all ρ 0 ), then there are thermodynamically isolated basins of state-space. In these cases, q  (12) is no longer directly valid. However, for any two initial density matrices ρ 0 and r 0 , such that r 0 has full rank relative to ρ 0 , it is still true that: Indeed q 0 , as defined in Eq. (51), is gauranteed to have full rank, and so Eq. (52) is valid if we set r 0 = q 0 . Alternatively, we can set r 0 = q [n] 0 if we properly restrict ρ 0 .
To proceed, we recognize that ρ 0 can be decomposed via Eq. (50) as: where ρ 0 projects ρ 0 onto the minimally dissipative basins, whereas ρ coh 0 describes the state's coherence between these basins.
Since q [n] 0 is, by definition, the minimally dissipative density matrix on its subspace (and, since it has full rank on that subspace), we have that: As an immediate consequence of their definition, the elements of {q Together with Eq. (55) this leads to: Eq. (58) can be seen as the quantum generalization of the classical result obtained recently in Ref. [32]. The classical version of this result is relevant when the minimally dissipative probability distribution (q 0 ) does not have full support. Dissipation on other 'islands' are then considered. Our derivation points out the nuances of physical assumptions that go into the classical result, and refines the notion of 'islands' (here referred to as 'basins') on a more solid physical grounding. The extension of these results to other optimization problems, as discussed in Ref. [32], is also discussed in one of our later appendices. Crucially, Eq. (58) generalizes the classical result-allowing ρ [n] 0 to exhibit quantum coherence relative to the minimally dissipative state q [n] 0 . In addition to the drop in Kullback-Leibler divergence on the minimally-dissipative eigenbasis, the drop in coherence also contributes to dissipation.
In the quantum regime, there is yet further opportunity for generalization, if we consider the possibility of coherence among the non-interacting basins of state-space. This is the case of non-zero inter-basin coherence: ρ coh 0 = 0. To address this more general case, we recognize that where ∆C ρ t (ρ t ) is the change in inter-basin coherence: from time t = 0 to time t = τ . Meanwhile, is the extra entropy flow due to inter-basin coherence.

APPENDIX D: CHANGE IN RELATIVE ENTROPY DECOMPOSES INTO CHANGE IN D KL S AND CHANGE IN COHERENCES
Our main result, Eq. (13), gave the extra dissipationwhen the system starts with the initial density matrix ρ 0 rather than the minimally-dissipative initial density matrix q 0 -in terms of the change in relative entropies between the two reduced density matrices over the course of the transformation: We now show how this can be split into a change in Kullback-Leibler divergences plus the change in the coherence on the minimally-dissipative eigenbasis.
The classical probability distribution that would be induced by projecting ρ t onto q t 's eigenbasis is: ρ t and P t only differ when ρ t is coherent on q t 's eigenbasis.
The actual state's coherence on q t 's eigenbasis is given by the 'relative entropy of coherence' [25]: Expanding the relative entropy between ρ t and q t at any time yields: where we used the simultaneously-diagonalized spectral representations of ln P t = s∈Λq t ln P t (s) |s s| and ln q t = s∈Λq t ln q t (s) |s s|, and where P t (s) ≡ s| P t |s = s| ρ t |s and q t (s) ≡ s| q t |s are the probability elements of the classical probability distributions P t and q t on the simplex defined by q t 's eigenstates.
(Similar decompositions of the quantum relative entropy appear in recent thermodynamic results of Refs. [33] and [34], although in a more limited context.) Thus, the difference in entropy production can be expressed as: as in Eq. (16) of the main text.
In the classical limit, where there are no coherences, we recover the classical result obtained by Wolpert and Kolchin-sky in Ref. [20]: From Eq. (71), we see that the quantum correction to the classical dissipation is exactly the change of coherence on the minimally-dissipative eigenbasis.

APPENDIX E: JUSTIFICATION FOR APPROACH TO THE GIBBS STATE UNDER WEAK COUPLING
Consider a system in constant energetic contact with a single thermal bath of inverse temperature β = 1 k B T . Suppose the system is undriven (i.e., x t = x t = x for all t, t > 0), so that it experiences a time-independent Hamiltonian H x . From Ref. [35], we can deduce that the system together with part of the thermal bath will together approach a stable passive state under the influence of the remainder of the thermal bath. For large baths, this stable passive state limits to the Gibbs state for the joint system. If we furthermore take the limit of very weak coupling, then this also yields the Gibbs state for the reduced system since e β(Hx⊗I b +Isys⊗H b ) = e βHx ⊗ e βH b . The system-bath interaction H int x can be treated as a small perturbation to the steady state with vanishing contribution in the limit of very weak coupling.
Hence, if this system starts out of equilibrium in state ρ 0 , then it will simply relax towards the canonical equilibrium state π x = e −βHx /Z, where Z is the canonical partition function of the system.
The case of strong coupling is more tricky because of the possibility of steady-state coherences in the system's energy eigenbasis [36]. Nevertheless, there are small quantum systems of significant interest that are rigorously shown to approach the Gibbs state as an attractor, even with strong interactions [37,38].

APPENDIX F: DISSIPATION, WORK, AND FREE ENERGY
Time-dependent control implies work and, in the thermodynamics of computation, entropy production is typically proportional to the dissipated work [4,30]. This appendix relates these quantities.
Since we allow for arbitrarily strong interactions between system and baths, some familiar thermodynamic equa-tions must be revised in recognition of interaction energies. Most of these revisions have already been thought through carefully in Ref. [23]. In this appendix, we spell out some of the general relationships among entropy production, heat, work, dissipated work, nonequilibrium free energy, and so on. This allows our results to be reinterpreted in terms of the various thermodynamic quantities.
Work is the amount of energy pumped into the universe of discourse by the time-varying Hamiltonian. It is the total change in energy of the system and baths: Subtracting the heat yields: which is the First Law of Thermodynamics if the interaction energy is treated as part of the system's energy.
If there is a single grand canonical bath at temperature T , then entropy production is related to the dissipated work and the nonequilibrium free energy. In that case, the dissipated work is: We see that the dissipated work is the work beyond the changes in nonequilibrium free energy and interaction energy. Any work not stored in free energy or interaction energy has been dissipated.
The nonequilibrium free energy always satisfies the familiar relation F t = U t − T S t : where the nonequilibrium addition to free energy is: π xt = e −βHx t /Z xt is the Gibbs state induced by the instantaneous control, and F eq xt = −k B T ln Z xt is the equilibrium free energy of the system, which utilizes the partition function Z xt = tr(e −βHx t ).
Even if the interaction energy is large, we see that we recover the familiar thermodynamic relations, as long as there is negligible net change in interaction energy over the course of the protocol: tr(ρ tot

APPENDIX G: RELAXATION TO NESS
Suppose that the system is in constant contact with at least two different thermal or thermo-chemical baths. We may think, for example, of a stovetop pot of water which is hot at its base and cooler at its top surface. Such a setup famously allows for the existence of nonequilibrium steady states (NESSs), like Rayleigh-Bénard convection [39]. Our concise result-only needing to compare ρ t and q t at times 0 and τ -may be quite useful to circumvent otherwise daunting thermodynamic analyses. 5 At a smaller scale, our results should allow new approaches to analyzing the thermodynamics of biomolecules like sodium-ion pumps or ATP-synthase that reliably break time symmetry in their NESSs via differences in chemical potentials across cellular membranes [41][42][43].
While there is not expected to be a general extremization principle for finding NESSs, the mere existence of minimally-dissipating initial states-or maximally-dissipating initial states 6 -implies the inprinciple-applicability of our results for the thermodynamic analyses of general NESSs. Caveats aside, there is an obvious opportunity to apply our results to systems with NESSs that do extremize entropy production, like certain steady states in the linear regime [14,21].

APPENDIX H: RELATION TO ERROR-DISSIPATION TRADEOFFS
Under control constraints-like time-symmetric drivingwhere fidelity costs significant dissipation [44], we find that q 0 may be forced to have eigenvalues of order and thus D ρ 0 q 0 can diverge as ln (1/ ). This is consistent with the generic error-dissipation tradeoff recently discovered for non-reciprocated computations in Ref. [44], but only explains the error-dissipation tradeoff for logically irreversible transitions like erasure. For logically reversible but nonreciprocal transitions, all initial distributions suffer the same error-dissipation tradeoff. In those cases, the error-dissipation tradeoff is not a consequence of the contraction of the relative entropy discussed here, but rather follows more generally from the theory laid out in Ref. [44].
With unrestricted control, arbitrarily high fidelity can be achieved with bounded dissipation.

APPENDIX I: RELATED OPTIMIZATION PROBLEMS
Our result appears superficially similar to a recent result by Kolchinsky et al. [31], which describes the initial-state dependence of nonequilibrium free energy gain. The results are nevertheless distinct since the initial state that leads to maximal free energy gain is typically not the same as the initial state that leads to minimal dissipation. Yet the similarity of the two results suggests a more general overarching result. Indeed, we have found a general theorem that contains these results as important cases.
Let the initial joint state of the universe be a product state of the system and environment: ρ tot 0 = ρ 0 ⊗ ρ env 0 , and suppose that the joint system evolves according to some unitary time evolution, such that the reduced state at time τ is given by: ρ τ = tr env Uρ 0 ⊗ ρ env 0 U † . We can consider any function of the initial density matrix: and its minimizer: Proof. If a(ρ) is an affine function, then it can be written as a(ρ) = (ρ) + c, where where (ρ) is a linear function of ρ and c is a constant. Representing the initial density matrix in an orthonormal basis as ρ 0 = j,k c j,k |j k|, and differentiating f (ρ 0 ) with respect to the matrix elements of ρ 0 , we find: = (|j k|) + tr |j k| ln ρ 0 − tr tr env U |j k| ⊗ ρ env 0 U † ln ρ τ .
If q 0 has a non-trivial nullspace, then Thm. 2 can be extended as done in App. C.
The classical limit of Thm. 2 is closely related to a result recently derived in Ref. [32,Thm. 1].
Another possibility is obtained if we simply let a(ρ) = 0. Then we find a new result about the initial-state dependence of entropy change: where q 0 ∈ argmin ρ0 S(ρ τ ) − S(ρ 0 ).
A simple example reveals that these optimization problems indeed have different solutions (i.e., different q 0 s). Consider a double-well energy landscape. The right well is raised in a very short duration τ in which the system cannot fully relax. The initial distribution that minimizes dissipation primarily occupies the left well. The initial distribution that maximizes free energy gain primarily occupies the right well.
We obtain further interesting results when a is a nonlinear function-which indicates the growth of other physically relevant quantities-and we will report on these elsewhere.

APPENDIX J: AN OBSERVATION ABOUT RELATIVE ENTROPIES
It is interesting to compare our main result with an expression for entropy production that can be derived (following Ref. [23]) from Eq. (4): It is an interesting mathematical observation that: when q 0 ∈ argmin ρ0 D ρ 0 ⊗ b π (b) U † x0:τ ρ τ ⊗ b π (b) U x0:τ . This seems related to the Pythagorean theorem of information geometry, which utilizes the information projection [45], but nevertheless appears to be distinct. It would be interesting to understand under what circumstances a similar mathematical relation holds. Given the prevalence of relative entropies in thermodynamics and quantum information, perhaps that understanding would lead to further physical insights.