PHYSICAL REVIEW X 8, 021011 (2018)

Fundamental Work Cost of Quantum Processes

Philippe Faist1,2,* and Renato Renner1 1Institute for Theoretical Physics, ETH Zurich 8093, Switzerland 2Institute for Quantum Information and Matter, Caltech, Pasadena California 91125, USA

(Received 5 September 2017; revised manuscript received 7 January 2018; published 10 April 2018)

Information-theoretic approaches provide a promising avenue for extending the laws of thermodynamics to the nanoscale. Here, we provide a general fundamental lower limit, valid for systems with an arbitrary Hamiltonian and in contact with any thermodynamic bath, on the work cost for the implementation of any logical process. This limit is given by a new information measure—the coherent relative entropy—which accounts for the Gibbs weight of each microstate. The coherent relative entropy enjoys a collection of natural properties justifying its interpretation as a measure of information and can be understood as a generalization of a quantum relative entropy difference. As an application, we show that the standard first and second laws of thermodynamics emerge from our microscopic picture in the macroscopic limit. Finally, our results have an impact on understanding the role of the observer in thermodynamics: Our approach may be applied at any level of knowledge—for instance, at the microscopic, mesoscopic, or macroscopic scales—thus providing a formulation of thermodynamics that is inherently relative to the observer. We obtain a precise criterion for when the laws of thermodynamics can be applied, thus making a step forward in determining the exact extent of the universality of thermodynamics and enabling a systematic treatment of Maxwell-demon-like situations.

DOI: 10.1103/PhysRevX.8.021011 Subject Areas: Quantum Information, Statistical Physics

I. INTRODUCTION understanding of nanoengines and information-driven thermodynamic devices [22–31], paving the way for exper- Thermodynamics enjoys an extraordinary universality— imental demonstrations [32–34]. applying to heat engines, chemical reactions, electromag- When studying the thermodynamics of small-scale netic radiation, and even to black holes. Thus, we are quantum systems, it is particularly relevant to define naturally led to further apply it to small-scale quantum the thermodynamic framework precisely. A customary systems. In such a context, the information content of a approach, the resource theory approach, is to investigate system plays a key role: Landauer’s principle states that the state transformations that are possible after imposing a logically irreversible information processing incurs an restriction on the types of elementary physical operations unavoidable thermodynamic cost [1]. Landauer’s principle that are allowed. Such frameworks have enabled us to has generated a new line of research in which information understand general conditions under which it is possible to and thermodynamic entropy are treated on an equal footing transform one state into another [35–42] and to study [2], in turn providing a resolution to the paradox of the erasure and work extraction in the single-shot regime Maxwell demon [3]. In the context of statistical mechanics, a [43–45]. Such results have been extended to the case where significant effort has also been made to elucidate the role of quantum side information is available [46,47], to situations the second law [4–9]. Statistical mechanics has further with multiple thermodynamic reservoirs [48–53], and to the provided important contributions to understanding the inter- case of a finite bath size [54–57]. The role of coherence and play between information and thermodynamics [10–18], catalysis has been underscored [58–66], the effect of with works studying the energy requirements of information – processing [19–21]. This has also led to an improved correlations studied [67 71], and the efficiency of nano- engines investigated [57,72–74]. Fully quantum fluctuation relations [75] and a second-law equality [76] have been *[email protected] derived, and further connections to the recoverability of quantum information have been exhibited [77]. Published by the American Physical Society under the terms of Furthermore, fully quantum state transformations were the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to characterized [78,79]. We refer to Ref. [80] for a more the author(s) and the published article’s title, journal citation, comprehensive review covering these approaches to quan- and DOI. tum information thermodynamics.

2160-3308=18=8(2)=021011(19) 021011-1 Published by the American Physical Society PHILIPPE FAIST and RENATO RENNER PHYS. REV. X 8, 021011 (2018)

−β Our main result is a fundamental limit to the work cost of where at inverse temperature β, the value e E is assigned any logical process implemented on a system with any to each energy level of energy E. Our main result then Hamiltonian and in contact with any type of thermody- quantifies how many pure qubits need to be invested, or namic reservoir. It accounts for the necessary changes in the how many pure qubits may be distilled, while carrying out a energy-level populations in the system, as well as for the specific logical process given as a completely positive, thermodynamic cost of resetting any information that needs trace-preserving map, subject to the restriction that the to be discarded by the logical process. It is valid for a single implementation must globally preserve the joint Γ operator instance of the process and ignores unlikely events, thus of the system and the battery. In this picture, the coherent capturing statistical fluctuations of the work cost. relative entropy intuitively measures the amount of infor- Our thermodynamic framework is specified by imposing mation “forgotten” by the logical process, conditioned on a restriction on the operations, which can be carried out the output of the process, and counted relative to the along with a battery system, allowing us to invest resources weights encoded in the Γ operator. to overcome this restriction. The restriction we consider Our framework can be applied to the macroscopic limit, here is to impose that the allowed operations must be to study transitions between thermodynamic states of a Gibbs-preserving maps, that is, mappings for which the large system. (For instance, an isolated gas in a box that is thermal state is a fixed point. This framework is a natural in a microcanonical state may undergo a process that brings generalization of the setup in Ref. [81] and has close ties to the gas to another microcanonical state of different energy resource theory approaches [36,38,41]. Gibbs-preserving and volume.) Remarkably, it turns out that the work cost of maps are the most generous set of physical evolutions that any mapping relating two thermodynamic states, as given can be allowed for free, in the sense that if any non-Gibbs- by the coherent relative entropy, is equal to the difference of preserving map is allowed for free, arbitrary work can be a potential evaluated on the input and the output state, extracted, rendering the framework trivial. Since in most regardless of the details of the logical process. For an existing thermodynamic frameworks the allowed free isolated system, we show that this potential is precisely the operations preserve the thermal state, our bound still holds thermodynamic entropy. By coupling the system to another in other standard settings such as the framework of thermal system that plays the role of a piston, i.e., that is capable of operations [38,41]. (However, if one considers catalytical reversibly furnishing work to the system, we recover the processes, more general transformations can be carried out, standard second law of thermodynamics relating the and hence additional care has to be taken in order to apply entropy change of the system to the dissipated heat. our framework, e.g., by including the catalyst explicitly as Our framework naturally treats thermodynamics as a part of the process [41,60,70,77].) As a battery system, we subjective theory, where a system can be described from the consider an information battery, that is, a memory register viewpoint of different observers. One may thus account for of qubits that are all individually either in a pure state or in a varying levels of knowledge about a quantum system. This maximally mixed state. The pure qubits are a resource that feature allows us to systematically analyze Maxwell- can be invested in order to implement logical processes that demon-like situations. Furthermore, we find a criterion that are not Gibbs preserving. certifies that the laws of thermodynamics hold in a coarse- Our main result is expressed in terms of a new grained picture. For instance, this criterion is not fulfilled in purely information-theoretic quantity, the coherent relative the case of the Maxwell demon, signaling that a naive entropy. The coherent relative entropy observes several application of the laws of thermodynamics to the gas may be natural properties, such as a data-processing inequality, disrupted by the presence of the demon. We hence obtain a invariance under isometries, and a chain rule, justifying its precise notion of when the laws of thermodynamics can be interpretation as an entropy measure. It is a generalization of applied, contributing to the long-standing open question of both the min- and max-relative entropy as well as the the exact extent of the universality of thermodynamics. conditional min and max entropy. In the asymptotic limit of The results presented in this paper have been, to a large many independent repetitions of the process (the i.i.d. limit), extent, reported in the recent work of one of the authors [85]. the coherent relative entropy converges to the difference of The remainder of the paper is structured as follows. In the usual quantum relative entropy of the input state and the Sec. II, we present the general setup in which our results are output state relative to the Gibbs state. Our quantity hence derived. In Sec. III, we explain our main result, the work adds structure to the collection of entropy measures forming cost of any process in contact with any type of reservoir the smooth entropy framework [82–84]. (Sec. III A); we then provide a collection of properties of In fact, our result may be phrased in purely information- our new entropy measure (Sec. III B), a study of a special theoretic terms, abstracting out physical notions such as class of states whose properties make them suitable energy or temperature in an operator Γ, which may be “battery states” for storing extracted work (Sec. III C), interpreted as assigning abstract “weights” to individual a discussion of how the macroscopic laws of thermody- quantum states. In the case of a system in contact with a namics emerge from our microscopic considerations heat bath, these weights are simply the Gibbs weights, (Sec. III D), and an analysis of how to relate the views

021011-2 FUNDAMENTAL WORK COST OF QUANTUM PROCESSES PHYS. REV. X 8, 021011 (2018) of different observers in our framework (Sec. III E). As mentioned above, in the case of a system S with Section IV concludes with a discussion and an outlook. Hamiltonian HS in contact with a single heat bath at inverse temperature β, we essentially recover the usual model of −β Γ HS II. FRAMEWORK OF RESTRICTED Gibbs-preserving maps by setting ¼ e . In the case of OPERATIONS multiple conserved charges such as a Hamiltonian HS, number operator NS, etc., we recover the relevant Gibbs- −β −μ Consider a system S described by a Hamiltonian HS.In preserving-maps model by setting Γ ¼ e ðHS NSþÞ,with the framework of Gibbs-preserving maps, an operation the corresponding chemical potentials, as expected; further- −β Φð·Þ is forbidden if it does not satisfy Φðe HS =ZÞ¼ more, the physical charges do not have to commute [52,53]. −β e HS =Z, where β is a given fixed inverse temperature and Our framework is designed to be as tolerant as possible −β Z ¼ tr½e HS . In other words, Φð·Þ is forbidden if it does (to the extent that our allowed operations are ultimately a not preserve the thermal state. Now, observe that the set of quantum channels), so as to result in the strongest condition on Φð·Þ depends on β and HS only via the possible fundamental limit. We start with this observation thermal state, so we can rewrite the condition in a more in the case of thermodynamics with a single heat bath: If we general, but abstract, way as follows: An operation Φð·Þ is allow any physical evolution for free that does not preserve forbidden if it does not preserve some given fixed operator the thermal state, then we may create an arbitrary number of Γ, that is, if it does not satisfy ΦðΓÞ¼Γ. We trivially copies of a nonequilibrium quantum state for free; however, −β recover Gibbs-preserving maps by setting Γ ¼ e HS .For this renders our theory trivial since usual thermodynamical technical reasons and for convenience, we choose to loosen models allow us to extract work from many copies of a the condition on Φ from being trace preserving to being nonequilibrium state. Accordingly, quantum thermody- trace nonincreasing; correspondingly, we only require that namics models that can be written as a set of allowed ΦðΓÞ ≤ Γ, instead of demanding strict equality. By enlarg- physical maps (such as thermal operations) necessarily ing the class of allowed operations, we can only obtain a have the Gibbs state as a fixed point, ensuring that our more general bound. The advantage of this abstract version fundamental limit applies for those models as well. We note of the Gibbs-preserving-maps model is that our framework that models in which catalysis is permitted allow for more and its corresponding results may be potentially applied to general state transformations [41,60,70,77], exploiting the any setting where a restriction of the form ΦðΓÞ ≤ Γ fact that, for a forbidden transition σ ↛ ρ, there might exist applies, for a given Γ, which does not necessarily have some state ζ such that σ ⊗ ζ → ρ ⊗ ζ (where ζ may be to be related to a Gibbs state. The way Γ should be defined chosen suitably depending on σ and ρ). In order to apply is determined by which restriction of the form ΦðΓÞ ≤ Γ our framework in such a context, we can consider the makes sense to require in the particular setting considered. catalyst explicitly. For instance, in the context of catalytic Finally, it proves convenient to consider non-normalized Γ thermal operations [41], after the catalyst has been included operators (this becomes especially relevant if we consider in the picture, the physical evolution that is applied is a different input and output systems). For instance, in the thermal operation and thus has to be Gibbs preserving. case of a system with Hamiltonian H in contact with a heat Ultimately, the correct choice of framework depends on the bath at inverse temperature β, the trace of Γ ¼ e−βH underlying physical model: For instance, in a macroscopic actually encodes the canonical partition function of the isolated gas, the whole system evolves according to an system. energy-preserving unitary, and under suitable independ- Our framework is defined in its full generality as follows. ence assumptions, the evolution of an individual particle is To each system S corresponds an operator ΓS, which may well modeled by a thermal operation; however, other be any positive semidefinite operator. We then define as situations might warrant the inclusion of a catalyst, for free operations those completely positive, trace-nonincreas- instance, in a paranoid adversarial setting in which an ing maps ΦA→B, mapping operators on a system A to eavesdropper may manipulate a thermodynamic system. In operators on another system B, which satisfy the first case, our ultimate limits apply straightforwardly, whereas in the second, one would need to include the catalyst explicitly. Φ Γ ≤ Γ A→Bð AÞ B: ð1Þ Work storage systems are often modeled explicitly but are mostly equivalent in terms of how they account for One may think of the Γ operator as assigning to each state work [2,38,39,58,81,86]. Among these, the information in a certain basis a “weight” characterizing how “useless” it battery is easily generalized to our abstract setting. An Γ is. As a convention, if ΓS has eigenvalues equal to zero, information battery is a register A of n qubits whose −βHA then the corresponding eigenstates are considered to be operator is ΓA ¼ 1A. (If ΓA ¼ e for an inverse temper- impossible to prepare—these states will never be observed. ature β and a Hamiltonian HA, the requirement that In the following, a map obeying Eq. (1) will be referred to ΓA ¼ 1A is fulfilled by choosing the completely degenerate as a Γ-subpreserving map. Hamiltonian HA ¼ 0.) The register starts in a state where λ1

021011-3 PHILIPPE FAIST and RENATO RENNER PHYS. REV. X 8, 021011 (2018) qubits are maximally mixed and n − λ1 qubits are in a pure state. In the final state, we require that λ2 qubits are maximally mixed and n − λ2 are in a pure state. The difference λ ¼ λ1 − λ2 is the number of pure qubits extracted or “distilled.” In this way, we may invest a number of pure qubits in order to enable a process that is not a free operation, or we may try to extract pure qubits from a process that is already a free operation. Depending on the physical setup, the λ pure battery qubits can themselves be converted explicitly to some physical resource, such as mechanical work. In the case where we have access to a single heat bath at temperature T, a pure qubit can be reversibly converted to and from kT ln 2 work using a Szilárd engine [22], where k is Boltzmann’s E constant; thus, a process from which we can extract λ pure FIG. 1. Implementation of a logical process (any quantum qubits is a process from which we can extract λ kT lnð2Þ process) using thermodynamic operations. The process acts on X and has output on X0, and is implemented by acting on the system work using the heat bath. More generally, we may replace and the battery with a joint Gibbs-preserving operation. The the information battery entirely by other battery models, battery starts with a depletion state λ1 and finishes with a such as corresponding generalizations to our framework of depletion state λ2. The overall extracted work is given by the “ ” the work bit (wit) [41], or the weight system [39,58]. difference λ1 − λ2. These work storage models are known to be equivalent [41]; the equivalence persists in our framework, with a “ ” λ when calculating the work cost of compressing an ideal gas, suitable generalization of the extracted resource . In the for instance, one ignores the exceedingly unlikely event presence of several physical conserved charges, and cor- λ where all gas particles conspire to hit against the piston at responding thermodynamic baths, the number of pure much greater force than on average, a situation that would qubits extracted acts as a common currency that allows us require more work for the compression but that happens to convert between the different resources. Hence, a λ with overwhelmingly negligible probability. For our pur- number of extracted pure qubits may be stored in different poses, we may optimize the zero-error work cost over states forms of physical batteries, corresponding to different that are ϵ approximations of the required state [81], which forms of work, such as chemical work [52,53]. Hence, λ is a standard approach in quantum information and cryp- the quantity should be thought of as a dimensionless tography [82,90]. value, expressed in number of qubits, characterizing the At this point, it is useful to introduce the notion of the “extracted resource value” of the logical process independ- process matrix associated with the pair ðEX→X0 ; σXÞ of the ently of which type of battery is actually used in the logical process and input state. First, we define a reference implementation, in the same spirit as the free entropy of system RX of the same dimension as X and choose some Ref. [52], and bearing some similarity to currencies in fixed bases fjki g and fjki g of X and R . Then, we general resource theories [87,88]. X RX X define the process matrix of the pair ðE → 0 ; σ Þ as the The main question we address may thus be reduced to X X X ρ 0 E 0 ⊗ σ σ bipartite quantum state X RX ¼ð X→X idRX Þðj ih jX∶R Þ, Γ Γ 0 ≥ 0 P X the following form (Fig. 1). Given operators X; X ,an 1=2 σ E where jσi ∶ ¼σ ð jki ⊗jki Þ. The process matrix input state X, and a logical process X→X0 (that is, a trace- X RX X X RX E nonincreasing, completely positive map), the task is to find corresponds to the Choi matrix of X→X0 , yet it is “ ” σ the maximum number of qubits that can be extracted, or the weighted by the input state X in the sense that the σ minimum number of qubits that need to be invested, in reference state is XRX instead of a maximally entangled order to implement the logical process on the given input state. The process matrix is in one-to-one correspondence state. Note that we require the correlations between the with the pair ðEX→X0 ; σXÞ except for the part of EX→X0 that E 0 σ ρ 0 input and the output to match those specified by X→X ,a acts outside the support of X; i.e., the specification of X RX condition that is not equivalent to just requiring that the uniquely determines σX as well as the logical process EX→X0 σ σ σ σ given input state X is transformed into the given output on the support of X. The reduced states X and RX of T state EX→X0 ðσXÞ. Equivalently, we require that the imple- σ are related by a partial transpose, σ σ . j iXRX RX ¼ X mentation acts as the process ðE → 0 ⊗ id Þ on a purified X X RX Intuitively, the reference system RX may be thought of state σ of the input, where id denotes the identity “ ” “ ” j iXRX RX as a mirror system, which remembers what the input process on RX. state to the process was. Finally, we ignore improbable events with total proba- As a further remark, one might be worried that bility ϵ, which is necessary in order to obtain meaningful the relaxation of the set of allowed operations from physical results [89]. Indeed, in textbook thermodynamics Γ-preserving and trace-preserving maps to Γ-subpreserving

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ˆ ϵ and trace-nonincreasing maps is too drastic. Indeed, while D 0 ðρ 0 kΓ ; Γ 0 Þ¼ max λ; ð2Þ X→X X RX X X −λ T ðΓ Þ≤2 Γ X X0 yielding a valid bound, the relaxed set of operations is T σ ≈ ρ ð XR Þ ϵ 0 unphysical, and we might thus obtain a looser bound than X X RX necessary. In fact, this is not the case. Rather, trace- where the optimization ranges over completely positive, nonincreasing, Γ-subpreserving processes are a technical trace-nonincreasing maps T → 0 . The notation “≈ϵ” sig- convenience, which allows for more flexibility in the X X nifies the proximity of the quantum states in terms of the characterization of what the process effectively does in purified distance, a distance measure derived from the the situations of interest to us while ignoring other fidelity of the quantum states related to the ability to irrelevant situations; yet, ultimately, we show that an distinguish the two states by a measurement [84,90,92], equivalent implementation can be carried out as a single which is closely related to the quantum angle, Bures trace-preserving, Γ-preserving map. For instance, consider distance, and infidelity distance measures [93,94]. a box separated into two equal-volume compartments, one The definition (2) is independent of which purification of which contains a single-particle gas (a setup known as a jσi is chosen on R , noting that ρ 0 also depends Szilárd engine [22]). The particle may be in one of two XRX X X RX on this choice. Furthermore, we use the shorthand states, jLi, jRi, representing the particle being located in ˆ ˆ ϵ¼0 D → 0 ðρ 0 kΓ ; Γ 0 Þ ≔ D 0 ðρ 0 kΓ ; Γ 0 Þ. either the left or right compartment. If the particle is located X X X RX X X X→X X RX X X in the left compartment, then work can be extracted by At this point, we may formulate our main contribution: attaching a piston to the separator and letting the gas Main Result. The optimal implementation of the E σ expand in contact with a heat bath. Yet, if we know the process X→X0 on the input state X, with free operations particle to be initially in the left compartment, it makes no acting jointly on the system X and an information battery, λ difference what the process would have done had the can extract a number optimal of pure qubits given by the particle been in the right compartment—that situation is coherent relative entropy, irrelevant. Hence, we may define the corresponding “effec- ˆ ϵ ” λ 0 ρ 0 Γ Γ 0 tive process as the trace-nonincreasing map, which maps optimal ¼ DX→X ð X RX k X; X Þ: ð3Þ jLi to the maximally mixed state (allowing us to extract λ 0 work) and which maps jRi to the zero vector. Evidently, the If optimal < , then the implementation needs to invest at −λ full actual physical implementation is a trace-preserving least optimal pure qubits. process, yet it is convenient to represent the “relevant part” The resources required to carry out the process, counted λ of this process using a trace-nonincreasing map. Crucially, in terms of optimal pure qubits, may be converted into both mappings have the same process matrix, given that the physical work. For instance, if we have access to a heat bath input state is jLi. This picture is justified on a formal level: at temperature T, we may convert each pure qubit into We show that any trace-nonincreasing, Γ-subpreserving kT lnð2Þ work and vice versa, and thus the work extracted ˜ map Φ can be dilated in the following way. There exists a by an optimal implementation of the process is trace-preserving, Γ-preserving map over an additional ϵ ρ 0 ˆ ancilla whose process matrix is as close to a given X R 2 0 ρ 0 Γ Γ 0 X W ¼ kT lnð Þ DX→X ð X RX k X; X Þ: ð4Þ as the process matrix of Φ˜ combined with a transition on the ancilla between two eigenstates of the Γ operator. In fact, it is not necessary to implement the process using (For technical details, we refer to the Supplemental the information battery at all, and the resources may be Material [91].) directly supplied by a variety of other battery models. The work can even be supplied by a macroscopic pistonlike system, as we will see later. III. RESULTS Here, we provide the main technical ingredients to understand the idea of the proof of our main result while A. Fundamental work cost of a process deferring details to the Supplemental Material [91]. Consider two systems X and X0 with corresponding A central step in our proof is a characterization of how Γ Γ operators X and X0 , respectively, as described above and much battery charge needs to be invested in order to exactly as imposed by the appropriate thermodynamic bath implement any completely positive, trace-nonincreasing σ [48,49,52,53]. We consider any input state X as well as map T → 0 . Such maps are those over which we optimize E X X any logical process X→X0 , i.e., any completely positive, in Eq. (2) to define the coherent relative entropy. The work R trace-preserving map. With a reference system X of the yield, or negative work cost, of performing T X→X0 with same dimension as X, which purifies the input state as Γ-subpreserving processes using an information battery is σ , the logical process and the input state jointly define “ Γ ” j iXRX given by how -subpreserving the process is. ρ 0 E 0 ⊗ σ T 0 the process matrix X RX ¼ð X→X idRX Þð XRX Þ. Proposition I. Let X→X be a completely positive, Our main result is phrased in terms of the coherent trace-nonincreasing map, and let y ∈ R. Then, the follow- relative entropy, defined as ing are equivalent:

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(a) The map T X→X0 satisfies 1. Elementary properties The coherent relative entropy obeys some trivial bounds. −y T X→X0 ðΓXÞ ≤ 2 ΓX0 : ð5Þ Specifically, − Γ − Γ−1 log2trð XÞ log2k X0 k∞ (b) For a large enough battery A (with ΓA ¼ 1A) and for λ λ ≥ 0 λ − λ ≤ ≤ ˆ ϵ ρ Γ Γ 1 − ϵ2 any 1, 2 such that 1 2 y, there exists a DX→X0 ð X0R k X; X0 Þþlog2ð Þ Γ Φ X trace-nonincreasing, -subpreserving map XA→X0A ≤ Γ−1 Γ log2k X k∞ þ log2trð X0 Þ: ð7Þ satisfying, for all ωX, These bounds have a natural interpretation in the context of −λ1 −λ2 ΦXA→X0A(ωX ⊗ð2 12λ1 Þ)¼T X→X0 ðωXÞ⊗ð2 12λ2 Þ; a single heat bath at inverse temperature β ¼ 1=ðkTÞ. The extracted work may never exceed an amount corresponding ð6Þ to starting in the highest energy level of the system and finishing in the Gibbs state; similarly, it may never be less −λ λ where 2 12λ denotes a uniform mixed state of rank 2 than the amount corresponding to starting in the Gibbs on system A. state and finishing in the highest excited energy level. Proposition I shows that if there is an allowed operation (A correction is added to account for additional work that in our framework which implements a given completely can be extracted by exploiting the ϵ accuracy tolerance.) positive, trace-nonincreasing map T exactly while charging Under scaling of the Γ operators, the coherent relative the battery by an amount λ, then the mapping must entropy simply acquires a constant shift: For any a; b > 0, necessarily satisfy T ðΓÞ ≤ 2−λΓ. Conversely, for any −λ trace-nonincreasing map T satisfying T ðΓÞ ≤ 2 Γ for ˆ ϵ ˆ ϵ b D 0 ðρ 0 kaΓ ;bΓ 0 Þ¼D 0 ðρ 0 kΓ ;Γ 0 Þþlog2 : some value λ, there exists an operation in our framework X→X X RX X X X→X X RX X X a acting on the system and a battery system which imple- ð8Þ ments T while charging the battery by some value λ. This operation is a trace-nonincreasing, Γ-subpreserving map In the case of a single heat bath at inverse temperature acting on the system and the battery. From this operation, β ¼ 1=ðkTÞ, this property simply corresponds to the fact we can then construct a fully Γ-preserving, trace-preserving that, if the Hamiltonians of the input and output systems map that implements T , as argued at the end of the previous are translated by some constant energy shifts, then the section. difference in the shifts should simply be accounted for in Our main result then exploits Proposition I in order to the work cost. Indeed, if HX → HX þ ΔEX and HX0 → −βΔEX −βΔE 0 answer the original question, that is, to find the optimal HX0 þ ΔEX0 , then ΓX → e ΓX, ΓX0 → e X ΓX0 and battery charge extraction when approximately implement- the optimal extracted work of a process, given by kT lnð2Þ ing a logical process E on an input state σ. In effect, one times the coherent relative entropy, has to be adjusted T −βΔE −βΔE needs to optimize the implementation cost over all maps according to Eq. (8) by kT lnð2Þlog2ðe X0 =e X Þ¼ whose process matrix is ϵ close to the required process ΔEX − ΔEX0 . matrix. This optimization corresponds precisely to the one carried out in the definition of the coherent relative entropy 2. Recovering known entropy measures in Eq. (2). (If σX is full rank and if ϵ ¼ 0, then necessarily T ¼ E; in general, however, a better candidate T may In special cases, we recover known results in single-shot be found.) quantum thermodynamics, reproducing existing entropy measures from the smooth entropy framework [82,84]. In the case of a system described by a trivial B. Coherent relative entropy and its properties Hamiltonian, the work cost of resetting the system to a ˆ ϵ 0 ρ 0 Γ Γ 0 The coherent relative entropy DX→X ð X RX k X; X Þ fixed pure state is given by the max entropy [43], a measure defined in Eq. (2) intuitively measures the amount of that characterizes data compression or information recon- information discarded during the process, relative to the ciliation [95]; similarly, preparing a state from a pure state weights represented in ΓX and ΓX0 . It ignores unlikely allows us to extract an amount of work given by the min events of total probability ϵ, a parameter that can be chosen entropy of the state, a measure that characterizes the freely. Its interpretation as a measure of information is amount of uniform randomness that can be extracted from justified by the collection of properties it satisfies, which the state. These results turn out to be special cases of are natural for such measures, and since it reproduces considering the work cost of any arbitrary quantum process known results in special cases. We provide an overview of for systems with a trivial Hamiltonian [81], which is given the properties of this quantity here and refer to the by the conditional max entropy of the discarded informa- Supplemental Material for the technical details [91]. tion conditioned on the output of the process:

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ˆ ϵ ˆ ϵ 0 ˆ ϵ 0 ρ 0 1 1 0 ≈− 4. Asymptotic equipartition DX→X ð X RX k X; X Þ HmaxðEjX Þ¼HminðEjRXÞ; ð9Þ An important property of the coherent relative entropy is where ρ 0 is a purification of ρ 0 and where its asymptotic behavior in the limit of many independent j iEX RX X RX ˆ ϵ 0 ˆ ϵ copies of the process (known as the i.i.d. limit). In this HmaxðEjX Þ and HminðEjRXÞ are the smooth conditional max entropy and min entropy that were introduced in regime, the coherent relative entropy converges to the Ref. [82] and are also known as the alternative conditional difference in the quantum relative entropies of the input max entropy and min entropy [96]. A precise meaning state to the output state, which is consistent with previous of the approximation in Eq. (9) is provided in the results in quantum thermodynamics [37,41]: Supplemental Material [91]. 1 ˆ ϵ ρ⊗n Γ⊗n Γ⊗n We recover more known results with an arbitrary lim D n→ 0n ð 0 k X ; 0 Þ n→∞ n X X X RX X Hamiltonian in contact with a heat bath by considering σ Γ − ρ Γ state formation and work extraction of a quantum state ¼ Dð Xk XÞ Dð X0 k X0 Þ; ð12Þ [38,44]. It is known that the work that can be extracted from recalling that σX is the input state of the process and ρX0 the a quantum state, or that is required to form a quantum state, resulting output state, and where ϵ is small and either kept is given by the min-relative entropy and the max-relative constant or taken to zero slower than exponentially in n. entropy, respectively; these single-shot relative entropies Crucially, the average work cost of performing a process in were introduced in Ref. [83] and are related to hypothesis the i.i.d. regime with Gibbs-preserving operations does not – testing [97 102]. We show that if the input or output system depend on the details of the process but only on the input is trivial, then and output states, as was already the case for systems described by a trivial Hamiltonian [81]. ˆ ϵ ρ Γ 1 ≈ ϵ ρ Γ DX→∅ð RX k X; Þ Dmin;0ð Xk XÞ; ð10aÞ 5. Miscellaneous properties ˆ ϵ ϵ We show a collection of further properties, including the D 0 ðρ 0 k1; Γ 0 Þ ≈−D ðρ 0 kΓ 0 Þ; ð10bÞ ∅→X X X max X X following: The coherent relative entropy is equal to zero for a pure process matrix, which corresponds to an identity matching the previously known results. We note that a mapping, for any input state and for ϵ ¼ 0; the smooth trivial system as output or input of a process is equivalent to coherent relative entropy can be bounded in both directions mapping to or from a pure, zero-energy eigenstate; this is as differences of known entropy measures; the coherent because the coherent relative entropy is insensitive to relative entropy does not depend on the details of the energy eigenstates (or more generally, eigenstates of the process if the input state is of the form ΓX=trðΓXÞ (e.g., a Γ operator) that have no overlap with the corresponding Gibbs state), and it reduces, in this case, to a difference of input or output state. input and output relative entropies and hence only depends on the output of the process. 3. Data-processing inequality and chain rule C. Battery states and robustness to smoothing The coherent relative entropy satisfies a data-processing inequality: If an additional channel is applied to the output, Previous work has already shown the equivalence of mapping the Gibbs weights to other Gibbs weights, then the several battery models known in the literature [41], notably “ ” coherent relative entropy may only increase. In other the information battery, the wit [38,41], and the weight system [39,76]. Our framework allows us to make this words, for any channel F 0→ 00 , X X equivalence manifest, by singling out a class of states on ˆ ϵ any system for which the system can act as a battery. These D 0 ρ 0 Γ ; Γ 0 X→X ð X RX k X X Þ states exhibit the property that they are reversibly inter- ˆ ϵ — ≤ 00 (F 0 00 ρ 0 Γ F 0 00 Γ 0 ) convertible (as in Ref. [103]) the resources invested in a DX→X X →X ð X RX Þk X; X →X ð X Þ : ð11Þ transition from one battery state to another can be recovered entirely and deterministically by carrying out the reverse Intuitively, this holds because the final state after the transition. application of F X0→X00 is less valuable as it is closer to For any system W with a corresponding ΓW, we consider the Gibbs state, and hence more work can be extracted by as battery states those states of the form the optimal process realizing the total operation X → X00. The coherent relative entropy also obeys a natural chain Γ τ P WP rule: The work extracted during two consecutive processes ðPÞ¼ ; ð13Þ trðPΓWÞ may only be less than an optimal implementation of the total effective process. We refer to the Supplemental where P is a projector such that ½P; ΓW¼0. In the Material [91] for a technically precise formulation. presence of a single heat bath at inverse temperature β,

021011-7 PHILIPPE FAIST and RENATO RENNER PHYS. REV. X 8, 021011 (2018) this class of states includes, for instance, individual energy include slight imperfections on the battery for any battery eigenstates or also maximally mixed states on a subspace of system: an energy eigenspace. We define the value of a particular 0 Γ battery state τðPÞ as ˆ ϵ trðPW WÞ 0 ρ 0 Γ Γ 0 − DX→X ð X RX k X; X Þ¼ max log2 ; ð16Þ W;P ;P0 ; Γ W W trðPW WÞ Φ → 0 Λ(τðPÞ) ¼ −log2trðPΓWÞ: ð14Þ XW X W τ where the optimization ranges over all battery systems W We require the system W to start in such a battery state ðPÞ Γ and to end in another such state τ P0 corresponding to with corresponding W, over all battery states corresponding ð Þ 0 Γ 0 Γ 0 0 0 Γ 0 to projectors PW, PW with ½PW; W¼½PW; W¼ , and another projector P with ½P ; W¼ . The following Φ ϵ proposition asserts that the system W can act as a battery over all free operations XW→X0W, which are an approxi- mation of a joint process XW → X0W, with a resulting enabling exactly the same state transitions on another ρ 0 process matrix on the system of interest given by X RX and a system S as an information battery with charge difference 0 0 transition on the battery from τðP Þ to τðP Þ [91]. λ1 − λ2 ¼ Λ(τðP Þ) − Λ(τðPÞ) (we again refer to the W W Supplemental Material for proof [91]): D. Emergence of macroscopic thermodynamics Proposition II. Let T X→X0 be a completely positive, trace-nonincreasing map, and let y ∈ R. Then, statements We now apply our general framework to the case of (a) and (b) in Proposition I are further equivalent to the macroscopic systems and recover the standard laws of following: thermodynamics as emergent from our model. On one (c) For any quantum system W with corresponding ΓW, hand, the goal of this section is to show that our framework 0 0 and for any projectors P, P satisfying ½P;ΓW¼½P ;ΓW¼0 behaves as expected in the macroscopic limit, further such that Λ(τðP0Þ) − Λ(τðPÞ) ≤ y, there exists a justifying it as a model for thermodynamics. On the other Γ-subpreserving, trace-nonincreasing map ΦXW→X0W such hand, the arguments presented here reinforce the picture that for all ωX, of the macroscopic laws of thermodynamics as emergent from microscopic dynamics, in line with common 0 – ΦXW→X0W(ωX ⊗ τðPÞ) ¼ T X→X0 ðωXÞ ⊗ τðP Þ: ð15Þ knowledge and existing literature [37,41,104 107],by providing an alternative explanation of this emergence The information battery, the wit as well as the weight based on Γ-subpreserving maps. (In fact, this emergence system are themselves special cases of this general battery may be understood as defining the order relation in −λ i λ – system. Indeed, the states 2 12 i of the information battery Refs. [104,105,108 110] as the ordering induced by trans- λ Γ can be cast in the form (13), with P ¼ 12 i since Γ ¼ 1 for formation by -subpreserving maps.) the information battery; the corresponding value of the state is indeed Λ(τðPÞ) ¼ −λi. Similarly, in the case of the wit 1. General mechanism and of the weight system, and in the presence of a single The macroscopic theory of thermodynamics is recovered −βHW heat bath at inverse temperature β such that ΓW ¼ e , when it is possible to single out a class of states that obey a the relevant states are energy eigenstates jEiW, whose reversible interconversion property. More precisely, sup- … value is precisely their energy, up to a factor β: pose there are a class of states fτz1;z2; ;zm g specified by m Λ(τ ) β ðjEihEjWÞ ¼ E. The equivalence of these models parameters z1; …;z , and suppose there exists a potential m … Λ … τz1; ;zm is thereby manifest. ðz1; ;zmÞ such that for any pair of states X and 0 … 0 As can be expected, the battery states of the general z1; ;zm τ 0 from this class, we have, for any process matrix form τðPÞ are reversibly interconvertible, implying that for X ρ 0 mapping one state to the other, any process that maps τðPÞ to τðP0Þ on a system, the X RX coherent relative entropy is equal to the difference ˆ lnð2Þ D → 0 ðρ 0 kΓ ; Γ 0 Þ Λ(τðPÞ) − Λ(τðP0Þ). X X X RX X X 0 0 This general formulation enables us to prove an inter- ¼ Λðz1; …;zmÞ − Λðz1; …;zmÞ: ð17Þ esting property of these battery states—they are robust to small imperfections. Indeed, when implementing a process The ln(2) factor merely serves to change the units of the on a system S using a battery W, it makes no difference coherent relative entropy from bits, which is standard in whether one optimizes over ϵ approximations of the overall information theory, to nats, which will prove convenient to process on the joint system S ⊗ W or over ϵ approxima- recover the standard laws of thermodynamics. We call the tions on S, only with no imperfections on the battery state function Λðz1; …;zmÞ the “natural thermodynamic poten- (as the smooth coherent relative entropy is defined above). tial” corresponding to the physics encoded in the Γ … More precisely, we prove that the smooth coherent relative operators. In other words, the two states τz1; ;zm and 0 … 0 entropy is exactly the optimal difference in the charge state τz1; ;zm may be reversibly interconverted, as any work of the battery while capturing all implementations that invested when going in one direction may be recovered

021011-8 FUNDAMENTAL WORK COST OF QUANTUM PROCESSES PHYS. REV. X 8, 021011 (2018) when returning to the initial state, and this is irrespective of which precise logical process is effectively carried out during the transition. An obvious choice is a state of the same form as the battery states introduced above, which motivates recycling the same symbols τ and Λ.[Wehave set ϵ ¼ 0 in Eq. (17) because smoothing such battery-type states has no significant effect.] Suppose that the parameters are sufficiently well approximated by continuous values. This would typically be the case for a large system such as a macroscopic gas. FIG. 2. Macroscopic thermodynamics emerges from our frame- Consider an infinitesimal change of a state ðz1; …;z Þ → m work when singling out a set of states that can be parametrized by ðz1 þ dz1; …;z þ dz Þ. If there is a free operation m m continuous parameters to a good approximation and can be that can perform this transition, then, necessarily, reversibly interconverted into one another. We consider the case the coherent relative entropy is positive; hence, of a textbook thermodynamic gas confined in a box, with a piston Λ … ≤ Λ … ðz1 þ dz1; ;zm þ dzmÞ ðz1; ;zmÞ. Conversely, if capable of furnishing work. In this setting, we recover the usual the coherent relative entropy is positive, then there neces- second law of thermodynamics, dS ≥ δQ=T, relating the change sarily exists a free operation implementing the said in entropy, the dissipated heat, and the temperature. transition. We deduce that the infinitesimal transition z → z þ dz is possible with a free operation if and only if 2. Textbook thermodynamic gas dΛ ≤ 0: ð18Þ We proceed to recover the usual laws of thermodynamics in this fashion for a macroscopic isolated gas S composed This condition expresses the macroscopic second law of of many particles (Fig. 2). The Hamiltonian of the gas is thermodynamics, as we will see below. denoted by HðVÞ, where the volume V occupied by the gas We may define the generalized chemical potentials is a classical parameter of the Hamiltonian that determines, for instance, the width of a confining potential. We assume, ∂Λ for simplicity, that the number N of particles constituting μi ¼ ; ð19Þ ∂z … … the gas is kept at a fixed value throughout, restricting our i z1; ;zi−1;ziþ1; ;zm considerations to the corresponding subspace. ∂ ∂ where the notation ð f= xÞy;z denotes the partial derivative Let us first consider the case of an isolated gas at fixed with respect to x of a function f,asy and z are kept parameters E, V. In order to apply our framework, we must constant. We may then write the differential of Λ as identify the Γ operator, which encodes the relevant restric- X tions imposed by the physics of our system. Recall that our restriction is meant to explicitly forbid certain types of dΛ ¼ μidzi: ð20Þ processes, without worrying whether a nonforbidden oper- ation is achievable. Here, we assume that at fixed E, V, the The generalized potentials μi are often directly related to physical properties of the system in question, such as system is isolated and hence evolves unitarily. In particular, E;V ðVÞ temperature, pressure, or chemical potential. the projector PS onto the eigenspace of H correspond- Γ Under external constraints on the variables z1;z2; …;zm, ing to energy E is preserved. Hence, the operator we may ask what the “most useless thermodynamic state” characterizing the gas alone for fixed E, V can be taken as compatible with those conditions is. The answer is given by ΓE;V E;V minimizing the potential Λ subject to those constraints— S ¼ PS : ð22Þ this is a variational principle. For instance, if two systems with natural thermodynamic potentials Λ1ðz1; …;zmÞ and This is compatible with standard considerations in statis- 0 0 Λ2ðz1; …;zmÞ are put into contact under the constraints that tical mechanics, which identify the state of the gas in such 0 conditions as the maximally mixed state in the subspace for all i, zi þ zi must be kept constant (such as for extensive E;V variables in thermodynamics), then we may write dzi ¼ projected onto by PS (the microcanonical state), which − 0 Λ Λ Λ τE;V E;V E;V dzi and minimize ¼ 1 þ 2 by requiring that we denote by S ¼ PS =trðPS Þ. Indeed, at fixed E, V X on the control system, an allowed transformation may not 0 Λ μ − μ0 ¼ d ¼ ð i iÞdzi; ð21Þ change this state. Now, we would like to account for changes in E, V.Itis μ μ0 and we see that the minimum is attained when i ¼ i. convenient to introduce a physical control system C, which If the system is undergoing suitable thermalizing dynamics, plays the following roles: It stores the information about then its evolution will naturally converge towards that all the controlled external parameters of the state in which point. the gas was prepared—here, the parameters are E, V;

021011-9 PHILIPPE FAIST and RENATO RENNER PHYS. REV. X 8, 021011 (2018) furthermore, it provides the necessary physical constraints dΛC ¼ νede þ νxdx; ð26aÞ on the gas and physical resources necessary for trans- formations, taking on the role of a battery. In our case, the dΛS ¼ μEdE þ μVdV; ð26bÞ control system includes a piston that confines the gas to a volume V and is capable of furnishing the energy required with the coupling inducing the relations dE ¼ −de and to change the state of the gas. For concreteness, we imagine dV ¼ Adx. The force f exerted by the piston onto the that the piston is balanced by a weight, causing the piston to gas is given by f ∂e=∂x . Using Eq. (26a), we see ¼ð ÞΛC exert a force f on the gas. The force f may be tuned by −1 that de ¼ νe ðdΛC − νxdxÞ, and hence f ¼ −νx=νe. The varying the weight. The states of the control system are thermodynamic work provided by the piston is the je; xiC, where e is the energy stored in the control system mechanical work performed by the weight, and x the position of the piston. The energy e is the ν potential energy of the weight, and it must be equal to δ − x − W ¼ fdx¼ dx: ð27Þ e ¼ Etot E as enforced by total energy conservation, νe where Etot is the fixed total energy of the joint CS system. τe;x → τeþde;xþdx Furthermore, x determines the volume of the gas as Any operation mapping two states CS CS , which V ¼ A · x, where A is the surface of the piston. If the obeys our global restriction, i.e., which preserves the control system were isolated and not coupled to the gas, then operator Eq. (23), must obey Eq. (18) or, equivalently, Λ ≤− Λ the nonforbidden operations on the controlP system would be d S d C; hence, Γ0 those preserving the operator C ¼ e;xge;xje; xihe; xjC, Λ ≤−ν − ν ν − δ ν δ where ge;x encodes the relevant physics of the control d S ede xdx ¼ eðdE WÞ¼ e Q; ð28Þ system: It decreases as either e increases or x increases, where we have defined the change in energy of the gas that meaning that a state je; xiC cannot be brought to the state 0 0 0 0 is not due to thermodynamic work as heat: δQ dE − δW. je ;xiC with e >eor je; x iC with x >x. In other words, ¼ we do not forbid reducing the weight charge or lowering it. The temperature of the gas is defined as Tgas ¼ −1 −1 The coupling between the control system and the gas can ð∂S=∂EÞ ¼ −ðkμEÞ as in standard textbooks, as be enforced with a Γ operator of the form the conjugate variable corresponding to entropy. The X − control system also acts as a heat bath, so we define its Γ ⊗ E¼Etot e;V¼Ax CS ¼ ge;xje; xihe; xjC PS : ð23Þ temperature T as the temperature of a gas that it would be e;x “in equilibrium” with, in the sense that our variational Λ If the control system is the state je; xi , then any allowed principle is achieved. The potential CS attains its mini- C mum under the constraints dE −de and dV Adx if operation must preserve the operator ΓE;V for the corre- ¼ ¼ S 0 dΛ μ −ν dE μ A−1ν dV E E − e V Ax ¼ CS ¼ð E eÞ þð V þ xÞ , implying that sponding ¼ tot and ¼ . Furthermore, Eq. (23) μ ν − ν −1 accounts for the physics of the control system itself with the E ¼ e and hence T ¼ ðk eÞ . We may now write Eq. (28) in its more traditional form, coefficient ge;x. − τe;x ⊗ τE¼Etot e;V¼Ax The states CS ¼je; xihe; xjC S are of the δQ dS ≥ : ð29Þ form (13); hence, they are reversibly interconvertible as per T Eq. (17), and they are a valid class of states for our macroscopic description. The corresponding natural Our control system is in fact another example of a battery thermodynamic potential is given as per Eq. (14), system. Indeed, it can convert another form of a useful resource, mechanical work, into the equivalent of pure Λ Λ Λ − CSðe; xÞ¼ Cðe; xÞþ SðEtot e; AxÞ; ð24Þ qubits for enabling processes on the system, while still Λ − Λ working under the relevant global constraints such as where we have defined Cðe;xÞ¼ lnge;x and SðE; VÞ¼ conservation of energy. − E;V E;V Ω ln trðPS Þ. Observe that trPS ¼ SðE;VÞ is the micro- The thermodynamic gas illustrates a situation in which Λ canonical partition function, and hence SðE; VÞ is, up to the macroscopic second law of thermodynamics is recov- Boltzmann’s constant k and a minus sign, the quantity ered as emergent. Note that the argument can also be Ω SðE; VÞ¼k ln SðE; VÞ, which is known as the thermo- applied to a system with different relevant physical quan- dynamic entropy of the gas: tities, such as magnetic field and magnetization of a −1 medium. ΛSðE; VÞ¼−k SðE; VÞ: ð25Þ As the gas is macroscopic, we assume that the param- E. Observers in thermodynamics eters E, V are well approximated by continuous variables. It In standard thermodynamics, one describes systems is useful to define the conjugate variables to e, x and E, V from the macroscopic point of view. This point of view via the differentials of ΛC and ΛS: is usually assumed only implicitly, to the point that notions

021011-10 FUNDAMENTAL WORK COST OF QUANTUM PROCESSES PHYS. REV. X 8, 021011 (2018) such as thermal equilibrium or the thermodynamic entropy (a) (b) function are often thought of as objective properties of the system. Yet, a closer look reveals that they can be thought of as observer-dependent quantities, which can be extended to observers with different amounts of knowledge about the system [46,47,111]. This observation is at the core of a modern understanding of the Maxwell demon. The present section begins with a brief motivation, reviewing a variant of the Maxwell demon. Then, we show that our framework is well suited for describing different observers and that it provides a natural notion of coarse FIG. 3. The Maxwell demon concentrates all particles on one graining. Indeed, the framework itself, thanks to the side of the box by opening the trap door at appropriate times. abstraction provided by the Γ operator, is scale agnostic (a) A macroscopic observer describing only the gas sees its and can be applied consistently from any level of knowl- entropy decrease, in apparent violation of the macroscopic edge about the system. More precisely, we show how to observer’s idea of the second law of thermodynamics. (b) The relate two descriptions from the viewpoints of two observ- demon observes no entropy change, as the state of the gas is conditioned on his knowledge. By modeling his memory as an ers, where one observer sees a coarse-grained version of ’ explicit system, originally in a pure state, we may understand his another observer s knowledge. The coarse graining is given actions as simply correlating his memory with the state of the gas. by any completely positive, trace-preserving map. We In doing so, a macroscopic observer may be induced into define a sense in which we can carry out the reverse witnessing a violation of a macroscopic second law. If the demon transformation, where one recovers the fine-grained infor- wishes to operate cyclically, he needs to reset his memory register mation, given the coarse-grained information, with the help back to a pure state, which costs work according to Landauer’s of a recovery map. This allows us to relate the laws of principle [2,3]; any work he might have extracted using his thermodynamics in either observer’s picture, where by the scheme is paid back at this point. “laws of thermodynamics” in an observer’s picture, we mean that the evolution of the system is governed in their the entropy of the gas? It is now clear that the answer picture by Γ-subpreserving maps. This provides a precise depends on the observer. The macroscopic observer sees criterion that can guarantee, in a given setting, that the laws the gas with its usual macroscopic thermodynamic entropy, of thermodynamics hold in the coarse-grained picture or, while the demon has engineered a state where the gas has intuitively, that “no Maxwell-demon-type cheating” is zero entropy conditioned on the side information stored in happening. Namely, if the fine-grained picture has no more his memory—he knows all there is to know about the gas. information than what can be recovered from the coarse- Conceptually, the thermodynamic reason for this difference grained picture, then our framework may be applied is that the demon is able to extract work from the gas, consistently from either picture, with both observers whereas the macroscopic observer is not. Indeed, the agreeing on the class of possible processes. demon can exploit the side information stored in his Consider the variant of the Maxwell demon depicted in memory to design a perfect trap-door opening schedule, Fig. 3. A gas is enclosed in a box separated into two equal which, when executed, concentrates all the particles on one volume compartments, which communicate only through a side of the box. (This process can itself be thought of as small trap door controlled by a demon. The demon is able CNOT gates acting in the other direction.) With all particles to observe individual particles and activates the trap door at concentrated on one side of the box, the demon can now appropriate times, letting a single particle through each extract work by replacing the separator by a piston and time, in order to concentrate all particles on one side of the letting the gas expand isothermally. (Of course, the memory box. From a macroscopic perspective, and looking only at register is still littered with all the information about the gas, one observes an apparent entropy decrease as the the gas; resetting the register costs work according to gas now occupies a smaller volume. However, from a Landauer’s principle, which is where the demon pays back microscopic perspective, the demon is essentially trans- his extracted work if he wishes to operate cyclically [2,3].) ferring entropy from the gas into a memory register, which The above example shows that a fully general framework is initially in a pure state [2,3]. Consider in more detail the of thermodynamics should be universally applicable from following process: The demon performs a series of CNOT the point of view of any observer, accounting for any level gates using the gas degrees of freedom as controls and his of knowledge one might possess about a system. One also memory qubits as targets, which “replicates” the informa- expects that if an observer sees a violation of their laws of tion about the gas particles into his memory. Since this thermodynamics, while knowing that in a finer-grained process is unitary, it preserves the joint entropy of the picture the corresponding laws are obeyed, then they may memory and the gas. The result is a classically correlated attribute this effect to lack of knowledge about microscopic state between the memory register and the gas. So, what is degrees of freedom which the observed process exploits.

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In the following, we show that our framework displays In order to give a precise meaning to the above state- these desired properties. ments, it is necessary to specify how a state described by Consider two observers, Alice and Bob, who have Bob can be translated back to Alice’s picture. Indeed, there distinct degrees of knowledge about a system. We assume can be several possible states for Alice that are compatible that the system’s microscopic state space HA, which Alice with Bob’s state. We describe this “recovery process” using has access to, is transformed by a completely positive, a recovery map, which gives, in a sense, the “best guess” of F A→B H H trace-preserving map A→B to a state space B, which is what the state on A could be, given only knowledge of used by Bob to describe the situation (Fig. 4). For instance, Bob’s state on HB. More precisely, we define the state Alice might have access to individual position and transformation from Bob’s picture to Alice’s picture as the momenta of all the particles of a gas, while Bob only application of a completely positive, trace-preserving map RB→A RB→A ΓB ΓA has access to partial information given by macroscopic B→A ð·Þ, with the property that B→A ð BÞ¼ A , recall- physical quantities such as temperature, pressure, volume, ΓB F A→B ΓA ing that B ¼ A→B ð A Þ. This ensures that the completely etc. More generally, if the microscopic system can be useless state in Bob’s picture is mapped back to the H ⊗ H embedded in a bipartite system K N that stores, completely useless state in Alice’s picture. An example respectively, the macroscopic information (available to of a suitable recovery map is the Petz recovery map both Bob and Alice) and the microscopic information [77,112–116], defined as (available to Alice only), then Bob’s observations can be ’ H related to Alice s simply by tracing out the N system. RB→A ΓA 1=2F A←B† ΓB −1=2 ΓB −1=2 ΓA 1=2 Suppose that Alice observes some microscopic dynamics B→A ð·Þ¼ A A←B ð B ð·Þ B Þ A ; ð30Þ happening within HA and that this evolution is Γ-preserv- ΓA F A←B† F A→B ing with a particular operator A . How does this evolution where A←B is the adjoint of the superoperator A→B . appear to Bob? It turns out that for Bob, these maps are also The Petz recovery map is completely positive and trace Γ Γ RB→A ΓB ΓA -preserving maps, but they are relative to his operator, preserving, and satisfies B→A ð BÞ¼ A (assuming that ΓB F A→B ΓA ΓB which is simply given as B ¼ A→B ð A Þ, that is, by B is full rank). ’ Γ ’ EA transforming Alice s operator into Bob s picture. Hence, given a trace-nonincreasing mapping A in Conversely, a map that appears as ΓB preserving to Bob Alice’s picture, we define Bob’s description of the mapping is observed by Alice as being ΓA preserving. as the composed map of transforming into Alice’s picture, applying the map, and transforming back to Bob’s picture:

EB F A→B ∘ EA ∘ RB→A B ¼ A→B A B→A : ð31Þ

EA EA ΓA ≤ ΓA Our claim is the following: If A satisfies A ð A Þ A , EB EB ΓB ≤ ΓB then B satisfies Bð BÞ B. Conversely, if we are given EB ’ a trace-nonincreasing mapping B in Bob s picture, then this map is described in Alice’s picture as the composed map of transforming to Bob’s picture, applying the map, and transforming back: FIG. 4. Observers in thermodynamics. Alice has access to microscopic degrees of freedom of a gas, while Bob can only EA ¼ RB→A ∘ EB ∘ F A→B; ð32Þ observe its coarse macroscopic properties, such as its temperature A B→A B A→B T, volume V, and pressure p. Alice describes the evolution of the ΓA we assert that if EB ΓB ≤ ΓB, then EA ΓA ≤ ΓA. gas using Gibbs-preserving maps, with a Gibbs state A on the Bð BÞ B A ð A Þ A full state space of the many particles of the gas. On the other The proof of both claims is straightforward, using hand, Bob describes the gas using his own knowledge—for F A→B ΓA ΓB RB→A ΓB ΓA A→B ð A Þ¼ B and B→A ð BÞ¼ A . More generally, instance, the macroscopic variables T, V, p—which, in full these claims hold as well for any trace-nonincreasing, generality, we can represent as a quantum state in a state space A→B B→A completely positive maps F , R satisfying H , which is obtained by applying a given mapping F A→B · on A→B B→A B A→B ð Þ F A→B ΓA ≤ ΓB RB→A ΓB ≤ ΓA Alice’s state. (For instance, this map may trace out the inacces- A→B ð A Þ B and B→A ð BÞ A , in which case ΓB sible microscopic information.) States of the gas described by B does not have to be full rank. Bob may be transformed to Alice’s picture by applying a suitable The above provides a general criterion that is able to recovery map, such as the Petz map [77,112–116]. Then, Alice’s guarantee that the laws of thermodynamics in the coarse- ΓA ΓB ’ A -preserving maps appear to Bob as B-preserving maps, where grained picture are valid: If the state of the system in Alice s ’ ΓB ΓB F A→B ΓA Bob s B operator is taken to be B ¼ A→B ð A Þ. Conversely, picture is one that can be recovered from Bob using a fixed ΓB ’ operations that preserve B for Bob may be described by Alice as recovery map, then Alice s free operations correspond to ΓA ’ ’ preserving A . free operations in Bob s picture, and hence Alice slawsof

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’ ΓB thermodynamics indeed translate to Bob s idea of what the an effective mapping on B that is not B subpreserving. laws of thermodynamics are. Enter the Maxwell demon. A simple example is the relation of the microcanonical to Our framework hence allows us to systematically ana- the canonical ensemble. (This is also known as Gibbs lyze a variety of settings inspired by the Maxwell demon. rescaling, an essential tool to relate thermal operations to Returning to our example depicted in Fig. 3, we identify noisy operations [38,41,45].) If Alice describes unitary Alice as possessing a microscopic description of the gas dynamics within an energy eigenspace of the joint system and the demon, and Bob as the macroscopic observer. and a large heat bath, then Bob describes the dynamics of The demon, as described by Alice, can perform Gibbs- the system alone as Gibbs-preserving maps. Consider a preserving operations on the joint system of the gas S and system S and a heat bath R, with respective Hamiltonians the demon’s memory register M, which, for simplicity, we HS and HR and total Hamiltonian HSR ¼ HS þ HR. choose to have a completely degenerate Hamiltonian HM ¼ Suppose that Alice has microscopic access to the heat 0 and thus ΓM ¼ 1M. Bob, on the other hand, describes the bath and hence describes the situation using the state space gas alone using standard thermodynamic variables, say, A ¼ S ⊗ R. Assume that the global state and evolution are energy E, volume V, and number of particles, N. To relate constrained to unitaries within a subspace of fixed total both points of view, we write the gas system (including a energy E. This evolution is, in particular, Γ subpreserving if possible control system to fix macroscopic thermodynamic ΓA E E ⊗ we choose A ¼ PSR, where PSR is the projector onto the variables) as a bipartite system S ¼ K N with states of ⊗ τE;V;N τE;V;N eigenspace of HSR corresponding to the energy E.On the form jE; V; NihE; V; NjK N , where N is the the other hand, Bob only has access to the system B ¼ S. microcanonical state corresponding to the macroscopic F A→B ’ τE;V;N E;V;N Ω The mapping , which relates Alice s point of view to variables E, V, N.Wehave N ¼ PN = ðE; V; NÞ, Bob’s, simply traces out the heat bath R. Bob then describes where PE;V;N projects onto the subspace of the microscopic ΓA N the operator A as system corresponding to fixed E, V, N, and where the partition function is Ω E; V; N tr PE;V;N . Then, Bob’s X ð Þ¼ ½ N B A picture is obtained from Alice’s by disregarding the Γ ¼ tr ðΓ Þ¼ gðE − E ÞjE ;kihE ;kj ; ð33Þ S R SR S S S S memory register as well as the microscopic information, E ;k S F A→B which corresponds to the mapping KNM→PKð·Þ¼trMNð·Þ. Alice uses the description ΓA ¼ jE; V; Ni where gðE Þ is the degeneracy of the energy eigenspace of KNM E;V;N R E; V; N ⊗ PE;V;N ⊗ 1 (see previous section). Bob, the heat bath corresponding to the energy E and where the h jK N M R ΓB vectors fjE ;ki g are the energy eigenstates on S with a on the other hand,P describes the gas using K ¼ S S F A→B ΓA Ω possible degeneracy index k. Following standard argu- KNM→Kð KNMÞ¼dM ðE; V; NÞjE; V; NihE; V; NjK, ments in statistical mechanics, and as argued in Ref. [38], where dM is the dimension of the system M. Using the fact F A←B† ⊗ 1 we have, in typical situations and under mild assumptions, that KNM←Kð·Þ¼ð·Þ NM, the Petz recovery map −β A→B − ∝ ES F gðE ESÞ e , and we hence recover in Eq. (33) the corresponding to KNM→K is determined to be standard canonical form of the thermal state. In other 1 words, Bob describes the dynamics on S as maps that RB→A ⊗ 1 † ⊗ M K→KNMð·Þ¼ðRK→KN½ð·Þ NRK←KNÞ ; ð34Þ preserve the Gibbs state. dM The above reasoning can be seen as a rule for trans- forming one observer’s picture into another; it remains where we have defined the operator important to analyze the situation in the picture that X E;V;N accurately describes the state of knowledge of the input PN R → ¼ jE;V;NihE;V;Nj ⊗ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð35Þ state of the agent carrying out the operations. The pictures K KN K Ω E;V;N ðE;V;NÞ are equivalent when Alice’s state of knowledge of A is no more than what B can recover using the recovery map, i.e., Importantly, the recovery map applied to any state of the RB→A ρB when her input state is exactly of the form B→A ð BÞ, form jE; V; NiK gives ρB ’ where B is the state of the system in Bob s picture. ΓA RB→A However, not all actions that Alice can perform using A - K→KNMðjE; V; NihE; V; NjKÞ subpreserving maps must induce a ΓB-subpreserving effec- 1 B ⊗ τE;V;N ⊗ M ; tive map on B. Indeed, if Alice’s input state is more refined, ¼jE; V; NihE; V; NjK N ð36Þ dM i.e., if she has more fine-grained information about the microscopic initial state than what Bob can infer, then her i.e., Bob assigns a standard thermal state to all systems that actions might appear to Bob as violating his idea of the he cannot otherwise access. From Alice’s perspective (the ’ 0 second law of thermodynamics. In this case, Alice may demon s), the memory register M starts in a pure state j iM, ΓA indeed perform A -subpreserving operations that result in in order to store the future results from observations of the

021011-13 PHILIPPE FAIST and RENATO RENNER PHYS. REV. X 8, 021011 (2018) gas. On the other hand, Bob has no way to infer this some physical laws that imply the restriction that the state from his macroscopic information. Because of this, evolution must preserve (or subpreserve) a certain operator Alice can design processes that are perfectly Γ subpreserv- Γ, then purity may be invested to lift the restriction on any ing from her perspective but which can trick Bob into process, as quantified by the coherent relative entropy; thinking he is observing a violation of the second law (as depending on how Γ is defined, one may express this described in Fig. 3). Consider, for concreteness, the abstract resource in terms of a physical resource such as following procedure: Alice performs a unitary process mechanical work. Furthermore, if the states of interest of ⊗τE;V;N ⊗ 0 0 mapping the state jE;V;NihE;V;NjK N j ih jM our system form a class of states that happen to be 2 2 ⊗ τE;V=2;N ⊗ −11 reversibly interconvertible, the macroscopic laws of to jE; V= ;NihE; V= ;NjK N ðdM MÞ, where we assume that the system M has just the right dimension to thermodynamics emerge, along with the relevant thermo- dynamic potential. In a coarse-grained picture, the thermo- store all the entropy resulting from mapping a state τE;V;N to N dynamic laws apply as long as our thermodynamic coarse- τE;V=2;N the state N of lower rank [we assume, for simplicity, graining criterion is fulfilled, namely, if the fine-grained τE;V=2;N τE;V;N that the rank of N divides that of N , and thus state is not more informative than what can be recovered ΩðE; V; NÞ¼dMΩðE; V=2;NÞ]. Alice’s process is fully Γ from the coarse-grained information. ΓA preserving because it is unitary and commutes with KNM. The notion of macroscopic limit considered here is more However, from Bob’s perspective, the gas changed its state general than assuming that the state of the system is a 2 ρ⊗n from jE; V; NiK to jE; V= ;NiK, in a blatant violation of product state , where each particle or subsystem is i.i.d. his idea of the second law of thermodynamics. Of course, a While typical thermodynamic systems are indeed close to clever Bob would be led to infer that there exists some an i.i.d. state (for instance, the Gibbs state of many system (M) that has interacted with the gas and absorbed noninteracting particles is an i.i.d. state), we only rely the surplus entropy. The point is, however, that Bob can still on a notion of “thermodynamic states,” defined by their very well apply his laws of thermodynamics (in the form of ability to be interconverted reversibly and with certainty. the restriction imposed by Γ-subpreserving maps) as long Thermodynamic states may include arbitrary interaction as Alice does not “actively mess with him.” In other words, between the particles or, in fact, may even be defined on a any observer can consistently apply the laws of thermo- small system of a few particles. More precisely, our notion dynamics (in the form of our framework) from their of thermodynamic states coincides with our definition of Γ perspective, using the restriction of -subpreserving maps battery states and corresponds to a state that is of the form for appropriately chosen Γ operators as long as this Γ Γ Γ Γ P P=trðP Þ for a projector P that commutes with . These restriction indeed holds. A -subpreserving restriction states can be reversibly interconverted in our framework, inferred from coarse graining a finer Γ-subpreserving and usual statistical mechanical states are precisely of this restriction fails exactly when the finer-grained observer form. The thermodynamic states may be used as reference actively makes use of their privileged microscopic access. charge states of a battery system, in the sense that they A further example illustrating the necessity of treating enable the same processes. thermodynamics as an observer-dependent framework, The core of the framework is the Γ-subpreserving where our framework could be applied, is provided by restriction imposed on the free operations. The Γ operator Jaynes’ beautiful treatment of the Gibbs paradox [111]. encodes all the relevant physics of the system considered. The restriction may be due to any physical reason—for IV. DISCUSSION instance, by assuming that the evolution is modeled by One might think that thermodynamics, as a physical thermal operations on the microscopic level, or by other- theory in essence, would require physical concepts, such as wise justifying or assuming that the spontaneous dynamics energy or number of particles, to be built into the theory, as are thermalizing in an appropriate sense. Furthermore, Γ is done in usual textbooks. Our results align with the subpreservation may come about in any situation where one opposite view, where thermodynamics is a generic frame- or several conserved physical quantities are being work itself, agnostic of any physical quantities such as exchanged with a corresponding thermodynamic bath, in “energy,” which can be applied to different physical a natural generalization of thermal operations [49,52,53]. situations, in the same spirit as previously proposed Our framework is not limited to usual thermodynamics: approaches [104,105,110,117–119]. The physical proper- By considering the Γ operator as an abstract entity, all ties of the system, such as energy, temperature, or number considerations in our framework are of a purely quantum of particles, are accounted for in our framework only information theoretic nature and make no explicit reference through the abstract Γ operator. to any physical quantity. For instance, one can consider Our results provide an additional step in understanding purity as a resource and impose that operations subpreserve the core ingredients of thermodynamics and hence the the identity operator; our framework applies by taking extent of its universality. Our approach reveals the follow- Γ ¼ 1; in this way, one can recover the max entropy as the ing picture: Given any situation where the system obeys number of pure qubits required to perform data

021011-14 FUNDAMENTAL WORK COST OF QUANTUM PROCESSES PHYS. REV. X 8, 021011 (2018) compression of a given state. We might further expect either general Gibbs-preserving maps or time-covariant connections with single-shot notions of conditional mutual Gibbs-preserving maps [79]. As a condition on state information [69,120–122], which in the i.i.d. case can also transformations, it automatically provides an upper bound be expressed as a difference of quantum relative entropies. to the amount of work one can extract when implementing Our approach is also promising for calculating remainder a specific process, which, in particular, implements a terms in recovery of quantum information [77,115, specific state transformation. Furthermore, the way the 123–126]. Furthermore, being a Γ-subpreserving map is covariance constraint is enforced in Ref. [79] provides a a semidefinite constraint, and thus optimization problems promising approach for including the covariance constraint over free operations may often be formulated as semi- in our framework as well and tightening our fundamental definite programs, which exhibit a rich structure and can be bound in the context of operations, which are restricted to solved efficiently. be time covariant. Finally, the conditions of Ref. [79] may Although the goal of our paper is to derive a fundamental be used to prove the achievability of state transformations limitation on operations in quantum thermodynamics, one with a covariant mapping; one could expect a suitable can also ask the question of whether this limit can be generalization of both frameworks to simultaneously han- achieved within a physically well-motivated set of oper- dle possible symmetry constraints and logical processes as ations. Because our bound is given by an optimization over well as state transformations, and a tolerance against Gibbs-preserving maps, it is clear that there is one such unlikely events using ϵ approximations. map that will attain that bound (or get arbitrarily close). Finally, our framework can describe a system at any However, it is not clear under which conditions our bound degree of coarse graining, including intermediate scales can be approximately attained in a more practical or between the microscopic and macroscopic regimes. We can realistic regime such as thermal operations (possibly consider, for instance, a small-scale classical memory combined with additional resources), as is the case for a element that stores information using many electrons or system described by a fully degenerate Hamiltonian [81] or many spins (such as everyday hard drives): The electrons for classical systems [127]. may need to be treated thermodynamically but not the The question of achievability is related to coherence in system as a whole since we have control over the the context of thermodynamic transformations, an issue of information-bearing degrees of freedom on a relatively significant recent interest [58–63]. In particular, thermal small scale. Other such examples include Maxwell-demon- operations do not allow the generation of a coherent type scenarios, which our framework allows to treat superposition of energy levels, while this is allowed, to systematically. Our framework is also suitable for describ- some extent, by Gibbs-preserving maps, which are hence ing agents who possess a quantum memory containing not necessarily covariant under time translation [127].Our quantum side information about the system in question. In approach suggests a possible interpretation for why this is other words, we provide a self-contained framework of the case: With Γ-subpreserving operations, one requires no thermodynamics, which allows us to make the dependence assumption that the system in question is isolated—for on the observer explicit, underscoring the idea that thermo- instance, Γ could be the reduced state on one party of a joint dynamics is a theory that is relative to the observer [111]. Gibbs state of a strongly interacting bipartite system. Indeed, the example in Ref. [127] can be explained in this ACKNOWLEDGMENTS way (see also Sec. 4.4.4 of Ref. [85]). Still, the question of whether Gibbs-preserving maps may be implemented We are grateful to Mario Berta, Fernando Brandão, approximately using a more practical framework, such Fr´ed´eric Dupuis, Lea Krämer Gabriel, David Jennings, as thermal operations (perhaps under certain conditions), and Jonathan Oppenheim for discussions. We acknowledge remains an open question. We note, though, that the contributions from the Swiss National Science Foundation coherence resources required in order to implement a (SNSF) via the NCCR QSIT as well as Project process can be determined using the techniques of No. 200020_165843. P. F. acknowledges support from the Ref. [66]. These general tools might thus clarify the precise SNSF through the Early PostDoc.Mobility Fellowship coherence requirements of implementing Gibbs-preserving No. P2EZP2_165239 hosted by the Institute for Quantum maps with covariant operations. In a similar vein, one could Information and Matter (IQIM) at Caltech, as well as from study the effect of catalysis in our framework [60,70,128], the National Science Foundation. presumably in the context of state transitions rather than logical processes. A closer study of this type of situation is expected to reveal connections with smoothed, generalized, free energies [129] and the notion of approximate majo- [1] R. Landauer, Irreversibility and Heat Generation in the rization [130]. Furthermore, we expect tight connections Computing Process, IBM J. Res. Dev. 5, 183 (1961). with recent results, providing a complete set of entropic [2] C. H. Bennett, The Thermodynamics of Computation— conditions for fully quantum state transformations under A Review, Int. J. Theor. Phys. 21, 905 (1982).

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