Supplementary Information for “ Correlations in quantum thermodynamics : Heat , work , and entropy production ”

Here we report the details of the examples. Details of example I State of the system Here we obtain the exact state of the total system up to the second order in the interaction coupling λ . After calculating the interaction-picture Hamiltonian H̃ ) int (τ) =U † 0 (τ)H (λ ) int U0(τ) and the corresponding evolution operator Ũλ (τ) = Te −i ∫ τ 0 dsH̃ (λ ) int (s), one can read the state of the combined system from ρ ) SB (τ) =U0(τ)Ũλ (τ)ρSB(0)Ũ † λ (τ)U † 0 (τ) (1) as ρ ) S (τ) =ρ (0) S (τ)+λ 2 { σ+ρ (0) S (τ)σ−∑ k | fk| |η(ω0,ωk,τ)| n(ωk,β )+ +σ−ρ (0) S (τ)σ+∑ k | fk| |η(ω0,ωk,τ)| ( n(ωk,β )+1 ) −∑ k | fk| ( ξ (ω0,ωk,ωk,τ)ρ (0) S (τ)σ+σ−+ξ (ω0,ωk,ωk,τ)σ+σ−ρ (0) S (τ) )( n(ωk,β )+1 ) −∑ k | fk| ( ξ (ω0,ωk,ωk,τ)ρ (0) S (τ)σ−σ++ξ (ω0,ωk,ωk,τ)σ−σ+ ρ (0) S (τ) ) n(ωk,β ) } +O(λ 3), (2) and similarly for bath B, ρ ) B (τ) =ρ β B + iλ ( Tr[ρS(0)σ+]e0 ∑ k f ∗ k η (ω0,ωk,τ) [ρ β B ,ak]+Tr[ρS(0)σ−]e −iω0τ ∑ k fk η(ω0,ωk,τ) [ρ β B ,a † k ] ) +λ 2 ( Tr[ρS(0)σ−σ+]∑ kk′ { f ∗ k fk′ η (ω0,ωk,τ)η(ω0,ωk′ ,τ)ak ρ β B a † k′ − fk′ f ∗ k ξ (ω0,ωk′ ,ωk,τ)ekk′ ) a † k′ak ρ β B − f ∗ k′ fkξ (ω0,ωk′ ,ωk,τ)e −i(ωk−ωk′ )τ ρ B a † kak′ } +Tr[ρS(0)σ+σ−]∑ kk′ { fk f ∗ k′ η(ω0,ωk,τ)η (ω0,ωk′ ,τ)a † k ρ β B ak′ − f ∗ k′ fkξ (ω0,ωk′ ,ωk,τ)e −iτ(ωk−ωk′ ) ak′a † k ρ β B − fk′ f ∗ k ξ (ω0,ωk′ ,ωk,τ)ekk′ )τ ρ β B aka † k′ }) +O(λ 3), (3)

As a preliminary and necessary step toward investigating heat and work exchanges between two interacting systems S and B, one needs to unambiguously assign to the two parties a percentage of the interaction energy depending on the state of the compound system. However, due to SB correlations, there will always be part of the interaction energy that belongs to both S and B together. In thermodynamic terms, extracting this part of the energy would require accessibility of the total system. Thus, we distinguish three contributions to the total internal energy of SB: one accessible only through S, the other one only through B, and the last one only through SB (as a whole) via the SB correlations. We call this latter contribution to the internal energy the binding energy. Certainly, although (in the case of time-independent total Hamiltonian) the total internal energy remains constant in time, that of either S or B varies because they interact and thus exchange work and heat.
In a recent publication 5 , the internal energy of an open quantum system has been defined as the energy which is accessible through measurements in a fixed "local effective measurement basis" 6 , and the definitions of work and heat suggested by considering the ability of the energy changes in altering the von Neumann entropy; heat is the energy flux that may change the entropy whereas work is the part of the energy change that keeps entropy intact. In contrast, in our formalism the internal energy associated with each subsystem is defined as the energy which is locally accessible in each individual subsystem by means of arbitrary local measurements. Although it has been known in the literature that correlations play a role in heat exchange, this fact has not been shown explicitly thus far. In the following, we provide explicit relations that exhibit the role played by correlations in heat, work, and entropy exchange between constituents of a bipartite system.
The structure of this paper is as follows. First we lay out the framework to define basic thermodynamic properties such as heat and work, and show a first law governing their mutual transformations. The next section deals with finite and infinitesimal versions of a possible formulation of the second law of thermodynamics. We illustrate our formalism through two examples. The paper is concluded by a summary.

First law of thermodynamics in the presence of interactions and correlations
We consider a closed quantum system SB consisting of two interacting quantum systems: the system of interest S and its bath or environment B-with no restrictions on the dimensionality of S and B. The state of SB is described by the density matrix ρ SB (τ), evolving under a total time-independent Hamiltonian The internal energy of the total system is the mean value of the total Hamiltonian with respect to the time-evolving state, SB tot t ot , and is thus constant in time since the dynamics of the total system is governed by the Schrödinger equation We assume ħ ≡ 1 throughout the paper.
In thermodynamics, infinitesimal variations of the internal energy of a system occur because of infinitesimal exchanges of heat d and/or work d between the system and the environment. The quantum mechanical counterparts of infinitesimal heat and work exchanges in a system with state ρ(τ) and time-dependent Hamiltonian H(τ) are given by 7 where "d" denotes a time differential, while  d and  d are in general inexact differentials. With these definitions, we have the following quantum version of the first law of thermodynamics for the internal energy In the case of a compound, isolated system SB, the states of the constituent subsystems are obtained by partial tracing ρ S,B (τ) = Tr B,S [ρ SB (τ)]; thus from equation (2) we have When there are correlations between the system and the environment, we can write

SB S B
where χ measures all correlations (classical or quantum). Replacing this decomposition into equations (6) and (7) yields is a nonlocal operator, that is, it cannot be assigned exclusively to either of subsystems S and B alone. It thus may seem physically reasonable (and rewarding) to modify this Hamiltonian such that it becomes locally inaccessible by S and B. An evident advantage of this redefinition is that it allows us to divide the total internal energy of a composite system into local parts, which are associated to and accessible by each system, and a nonlocal part, which is attributed only to these systems as a wholewhereas it does not contribute to the (local) energies of each system. which is in general a nonzero scalar. One may remedy this in various ways. For example, a straightforward fix is to compensate for the nonzero scalar contribution of equation (14) by distributing it over the system and environment Hamiltonians through the real (but not necessarily positive) auxiliary parameters α S and α B = 1 − α S and hence defining the effective Hamiltonians in equation (12) can be replaced with an effective interaction Hamiltonian Note that the dynamical equations (9) and (10) an energy contribution remains, called binding energy, which can be naturally attributed to correlations χ as

S B tot
Note that only if the interaction and correlations between the two systems were negligible (which is usually assumed in classical thermodynamics)-namely only if H int ≈ 0 and ρ SB (τ) ≈ ρ S (τ) ⊗ ρ B (τ) at all times τ-we and the internal energy be additive.

Exchange of heat and work between two interacting systems
When two thermodynamical systems are independent, namely uncorrelated and noninteracting, the amount of the transferred heat/work from one system is equivalent to the heat/work received by the other system. In our context, the existence of interactions and correlations between S and B alters this picture for heat exchange, but Scientific RepoRts | 6:35568 | DOI: 10.1038/srep35568 not for the exchanged work. Indeed, inserting equation (15) into equation (4), the infinitesimal works performed by S and B are obtained as The work absorbed or released by system S, respectively B, each depends on the scalars α S,B , which is reminiscent to the non-gauge-invariance feature of work 9 . Nevertheless, we always have

S B
which is independent of α S,B , and looks like a form of the familiar classical law of action-reaction. Note that, unlike in refs 2 and 10 where it is the time dependence of the Hamiltonian through an external driving parameter that leads to work exchange, here the work exchange follows from the time dependence of the effective Hamiltonians which include time-dependent Lamb-shift-like corrections in equation (17). As a consequence, our formalism features that work exchange between two interacting subsystems is allowed even without an external driving.

S B
Hence, in our setting the binding energy is completely of the heat type. This can also be seen from the relations , which show that changes in the binding energy can only come from changes in the state correlations. Indeed, the heat balance equation (26) shows that (i) heat transfer is only due to interactions and correlations within the total system, in agreement with the result of refs 6 and 11, and (ii) that heat passing from one system to the other is paid for by varying the SB correlations that thus behave like a heat storage. That is, if correlations do not change and dχ(τ) = 0, then  = , in agreement with the standard textbook definition of "heat" in classical systems wherein no or only a negligibly-weak interaction between the system and the environment is assumed 12 .

Remark 2.
In our derivations thus far we have assumed that the Hamiltonian of the total system SB is time-independent. If we relax this condition and allow a time dependence (e.g., due to the action of some external agent on the total system), part of our relations will be modified as follows: . Equation (29) indicates that even in the time-dependent case correlations do not contribute in the exchange of work between the system and the environment. Since the dynamics of the compound system SB is generated by H tot (τ), it follows that Tr[dρ SB (τ)H tot (τ)] = 0 (i.e., the total system is thermally isolated), so that there are no heat exchanges and the only possibility for SB is to perform work because of the external driving due to the rest of the universe.

Second law of thermodynamics
According to the second law of thermodynamics, the entropy of a macroscopic closed system which is thermally isolated (in thermodynamics terminology) can only remain constant or increase in time 12,13 . However, the second law is not necessarily valid in nonequilibrium microscopic or even macroscopic systems [14][15][16][17][18][19] .
In the following we demonstrate the possible emergence of the second law of thermodynamics and the important role of system-bath correlations in this microscopic context.
In the case of a compound system SB, the subadditivity of the von Neumann entropy 20 (we set κ B ≡ 1 for the Boltzmann constant throughout the paper) implies that the mutual information is always nonnegative. Mutual information characterizes the amount of total correlations (both classical and quantum) shared by the two subsystems S and B 21,22 . Intuitively, if the correlations between S and B increases,  χ becomes larger.
Since we have assumed that the total system SB is closed, it evolves unitarily and its von Neumann entropy  SB (τ) does not change in time (even if its Hamiltonian depends on time). Hence, differentiating equation (31) yields Integrating both sides of this equation in the time interval [0, τ], with the assumption that the initial state of SB is uncorrelated (i.e.,  χ (0) = 0), leads to S B as obtained in ref. 23. This relation states that, as long as one observes subsystems S and B locally and their initial state is without any correlations, the sum of the total variations of the entropies of S and B is always nonnegative.
One can consider this property as a form of the second law of thermodynamics for the compound system SB.
Unlike in equilibrium thermodynamics, in a general nonequilibrium system "temperature" is not a well-defined quantity (see, e.g., refs 24 and 25 for some recent discussions). However, at fixed "volume" (V) and "number of particles" (N), one can introduce a time-dependent pseudo-temperature by means of the internal energy and the von Neumann entropy through which is somewhat reminiscent of the standard, equilibrium definition S U

Remark 3. In generic quantum systems, it is not always clear how to define V and N (or other relevant thermodynamic properties). Additionally, in thermodynamic equilibrium we deal with the partial derivative
rather than the ratio of two total derivatives ( S U τ τ (d /d )/(d /d )), which can be different quantities. Noting equations (15) and (16), the free parameter α S (and α B ) would also appear in the pseudo-temperature. In general then, one should not expect that the pseudo-temperature necessarily have definite relation with the equilibrium temperature, unless under certain conditions. Later in the examples we show explicitly how in special cases the pseudo-temperature may relate to the equilibrium temperature by appropriately fixing the scalar α S through thermodynamic properties of the system in question.
Adopting the concept of pseudo-temperature, one can associate (time-dependent) pseudo-temperatures T S,B (τ) with subsystems S and B and also a pseudo-temperature T χ (τ) with the binding energy. As a result, inserting equation (34) into equation (32) gives It then follows that, [equation (21)], the two subsystems must have the same instantaneous pseudo-temperature: T S (τ) = T B (τ). Another possibility is when We remark that a somewhat similar result in ref. 26 is akin to our general expression in equation (35) (of course here with pseudo-temperature instead of equilibrium temperature). This interesting relation yet again indicates the role of correlations; for energy transport to the bath (and similarly to the system) correlations are necessary, where in turn development of correlations ensues from interaction.

Remark 4. Under the same conditions, when the total Hamiltonian is time-dependent, equation (36) is modified to
Furthermore, from equations (5) and (34), it follows that The quantity Σ ∼ d S B , resembles the infinitesimal internal entropy production as defined in ref. 2, where the case of an externally driven system S has been discussed which is weakly coupled to a conservative heat bath B inducing a dissipative dynamics 8,[27][28][29] (in general explicitly time-dependent) of the Lindblad type. In this particular context, the infinitesimal entropy production is modified as the difference between the variation of the entropy  τ ( ) S and the entropy flux into or out of the system associated to the heat flux  τ d ( ) S divided by the (initial) temperature of the bath T (rather than T S (τ) as in equation (38)),

S S S
This expression can be interpreted as an internal entropy production for system S and its nonnegativity,

Remark 5. A simpler physical context is provided when there is no external driving for system S, namely its Hamiltonian H S (τ) = H S is time-independent, and so are the Lindblad generator
In such a case, the proof of the positivity of the entropy production follows from equation (40) becoming and from the monotonicity of the relative entropy under completely-positive, trace preserving dynamics 30 .
In the finite expression of the second law of thermodynamics (which follows from equation (33) in the absence of initial correlations between S and B), the heat bath B is taken explicitly and directly into account (though the term  ∆ B ). Rather, in the infinitesimal expression (42), the heat bath is indirectly accounted for by the fact that (i) the heat exchange occurs at the bath temperature, and (ii) that the dissipative reduced dynamics of system S is determined by the bath in the weak-coupling limit.
Notwithstanding these fundamental physical differences, it is still interesting to study to which extent the thermodynamical inequality cannot be both strictly positive in general. For example, in the case of the same instantaneous pseudo-temperatures, as when  τ = χ d ( ) 0 and T χ (τ) ≠ 0, from . In general, it is not true that the finite variation becomes nonnegative in the absence of initial correlations between S and B-unlike the case for the finite variations of the von Neumann entropies of the reduced states ρ S,B (τ).
One can argue that the infinitesimal quantities do not generically behave as expected from true thermodynamic quantities because the instantaneous pseudo-temperatures do not behave themselves as thermodynamic temperatures. This, however, does not exclude that, under certain conditions, proper thermodynamic patterns might emerge.
To alleviate the above situation, we can discern a better motivated notion of temperature by appealing to analogy with standard thermodynamics. In classical thermodynamics the relation holds for a system undergoing a quasistatic reversible transformation, whereas for a nonequilibrium process there is an extra term corresponding to the internal entropy production Σ , Scientific RepoRts | 6:35568 | DOI: 10.1038/srep35568 In this case the "temperature" is fixed by the external environment (bath) which is supposed to exchange heat always quasistatically (because of its short relaxation times), without changing its temperature. In our formalism, however, we treat the system and bath similarly. Thus we can extend equation (47) and identify an extended temperature and an entropy production for both system and bath and see how they compare at long times with expected thermodynamic temperatures. One way to do so is to explicitly compute d and d and next compare them to derive a physically meaningful extended temperature  τ ( ) as  Remark 6. Note that equation (48) defines both the extended temperature τ ( )  and the generalized entropy production dΣ (τ). Moreover, unlike the pseudo-temperature τ T( ), τ ( )  is by construction α S,B -independent because neither heat nor entropy depends on α S,B . In the following examples, we discuss both nonequilibrium temperatures T(τ) and τ ( )  by comparing them with the equilibrium temperature T (of the bath).

Examples
Here we study in detail two examples, in one of which thermalization occurs, whereas the other one does not exhibit this feature.
Example I: Thermalizing qubit. Consider a two-state system (e.g., a spin-1/2 particle or a two-level atom) interacting with a thermal environment, comprised of infinitely many modes at (initial) temperature T = 1/β, through the Jaynes-Cummings total Hamiltonian = Here σ x , σ y , and σ z = diag (1, − 1) are the Pauli operators, σ ± = σ x ± iσ y , and a k is the bosonic annihilation operator for mode k. Although this model is not exactly solvable, we can find the exact states of the system and bath up to any order in λ; see supplementary information for details of O(λ 3 ) calculations. A separate study of the dynamics of the subsystems in the damped Jaynes-Cummings model can be found in ref. 32.
In the weak-coupling, long-time, ω-continuum, Markovian limit (where λ → 0 and τ → ∞ such that λ 2 τ = const. and ), we can find the following Lindblad-type dynamical equation: is the Lamb-shift Hamiltonian,  denotes the Cauchy principal value, β is the inverse temperature of the bath, . It is evident from this solution that system S eventually thermalizes, as ρ S = (1/2)(I + r · σ) (here σ = (σ x , σ y , σ z )), and from equation (51) we have Additionally, in the Markovian limit, the energy of this system is obtained as As a result, This pseudo-temperature behaves well, i.e., exhibits thermalization, if there is no initial coherence (ρ 10 = 0, or equivalently, x(0) = y(0) = 0). In the Markovian regime we consider the thermal bath always in equilibrium (namely, ρ τ ρ ≈ β ( ) B B ), and as a consequence the effective energy of S reduces to (see equation (8)) For the bath thermodynamics, after some algebra we find that when τ → ∞ (up to O(λ 3 )) If we now consider x(τ), y(τ) and  τ ( ) evolving according to the dissipative thermalizing dynamics (56), we obtain x y , which agrees with the standard definition of the equilibrium temperature. Rather, the inverse pseudo-temperature τ which corresponds to inverting the function  τ ( ), finding  τ ( ), and computing the total derivative with respect to , (i.e., vanishing off-diagonal elements, ρ 10 = 0), it will not evolve in time, and because of equation (110) the bath will not experience any entropy change either;  τ = . λ ( ) const B ( )

Summary and Outlook
This paper highlights the role of correlations in the nonequilibrium thermodynamic behavior of generic bipartite interacting quantum systems. In this formulation, interesting relations emerge between correlations, on the one hand, and heat, work exchanges, as well as possible definitions of nonequilibrium temperatures of each subsystem, on the other hand. These relations may enable the extraction of desired thermodynamic properties by partially controlling or manipulating the underlying dynamics of the system. A notion of binding energy has been introduced which only depends on the interaction Hamiltonian and correlations of the total system state, whose variation has been shown to be only of the heat type. In addition, this energy has been shown not to be locally accessible by the subsystems, but it provides a heat transmission channel between the parties. In this sense, correlations act as a resource or storage for heat. We have also defined two notions of nonequilibrium temperatures for the subsystems and discussed their relevance in the thermodynamic equilibrium. We have also associated a nonequilibrium temperature with correlations. This temperature may enable one to obtain conditions such that the two subsystems have same nonequilibrium temperatures, which are generically different exactly because of correlations. These results have been illustrated in detail through two examples: a qubit in interaction with a thermalizing bath and a qubit interacting with a dephasing environment.
Our methodology may provide techniques and tools for employing quantum resources, such as manybody correlations and memory, to engineer thermodynamic processes, for example, to build efficient quantum heat engines, or shed light on our understanding of the role of correlations in biological processes in relation to, e.g., the efficiency of photosynthetic light-harvesting complexes 34 .