Exchange fluctuation theorems for strongly interacting quantum pumps

The unrivaled success of thermodynamics originates in the very nature of the theory. In contrast to microscopic descriptions, equilibrium thermodynamics formulates universally valid statements based on phenomenological observation. Its central quantity is the entropy, which, in a causal universe, can never decrease. However, in far from equilibrium situations, it is often required to reformulate some statements. This explains the importance and impact of the fluctuation theorems,^1–5 which can be understood as symmetry relations for the probability distribution of entropy production for any physical scenario in the universe.

Remarkably, these fluctuation theorems can be understood as generalizations of the second law of thermodynamics,⁶ and they can be shown to contain previous extensions of thermodynamics away from equilibrium, such as the linear response theory.^7,8 Different versions of these theorems have already been experimentally verified in different systems including biomolecular systems,⁹ nuclear magnetic resonance systems,¹⁰ trapped ion systems,^11–13 and nitrogen-vacancy (NV) centers in diamond.^14,15

One of the major conceptual building blocks of stochastic thermodynamics, a non-equilibrium reformulation of the classical scenario, is to recognize that thermodynamic quantities such as work, heat,¹⁶ and entropy production¹⁷ can be defined along single trajectories. Therefore, it is somewhat natural that the first focus was put on thermodynamic work since it depends only on the action of a controllable external agent.^3,4 Arguably, heat is a more interesting and complicated quantity, which is related to the uncontrolled forms of the energy exchange. Indeed, fluctuation theorems were also unveiled for heat exchanged between systems¹⁸ as well as more general “heat-like” quantities.^19–21 In the present Letter, we will study the fluctuation theorem for heat, or rather more generally any form of energy, exchanged between interacting quantum systems.

In quantum thermodynamics,²² the major problem is that classical trajectories become ill-defined. Hence, any thermodynamic quantity defined along a classical trajectory needs to be carefully re-defined. One of the most successful paradigms has been dubbed two-time measurement (TTM) scheme,^{12,18,23–37} in which thermodynamic quantities are determined from two projective measurements of the energy at the beginning and at the end of a process. While such an approach is more geared toward work in isolated quantum systems, it has also found application when considering heat in open quantum systems.^18,27,38,39

However, it has been argued that the TTM scheme is thermodynamically inconsistent since the projective energy measurements inevitably destroy quantum coherences.⁴⁰ To overcome this shortcoming of the TTM scheme, recent works proposed alternative paradigms, such as dynamic Bayesian networks,⁴¹ the Maggenau–Hill quasiprobability,⁴² and the one-time measurement (OTM) scheme.^43–47 Particularly, in the OTM scheme, the distribution of changes in internal energy is constructed by considering the expectation value of the energy conditioned on the initial energy measurement outcomes. This formalism avoids the second projective measurement, and, therefore, the thermodynamic contribution of quantum coherence or the correlations generated by the dynamics are naturally contained in the formalism.⁴⁸ Therefore, particularly in the strongly coupled systems, the effect of the quantum correlations is non-negligible, so that the OTM scheme naturally becomes the most appropriate paradigm.

In this letter, we employ the OTM scheme to derive a generalized quantum exchange fluctuation theorem for the multipartite case with arbitrary coupling strengths. As a consequence, we obtain a tighter bound on the net entropy change, which can be characterized by the quantum mutual information of a thermal state conditioned on the initial energy measurement. Our main results elucidate the role of quantum correlations in the heat exchange between arbitrary quantum systems.

This letter is organized as follows. In Sec. II, we first summarize the notions and notations which we use in this Letter. Then, in Sec. III, we discuss the heat distribution for the multipartite systems in the two-time measurement scheme, from which we derive the corresponding quantum exchange fluctuation theorem and the second law of thermodynamics and recover the bipartite case studied in Ref. 18. Third, in Sec. IV, we discuss the heat distribution for the multipartite systems in the one-time measurement scheme and derive the corresponding quantum exchange fluctuation theorem and second law of thermodynamics with a tighter lower bound, which is associated with the quantum mutual information of the conditional thermal state. Fourth, in Sec. V, we verify the tighter Clausius bound by employing an example of tripartite XY model. Finally, we conclude in Sec. VI.

For ease of notation and to avoid clutter in the formulas, we work in units for which the Boltzmann constant k_B and the reduced Planck constant $ℏ$ simply are $k B = ℏ = 1$⁠. We consider a d-dimensional multipartite system described by the composite Hilbert space $H ≡ ⊗ j = 1 n H j$⁠, where $H j$ denotes the Hilbert space of the jth subsystem. The corresponding, time-independent Hamiltonians are H_j and $H = ∑ j = 1 n H j$⁠. Here, H_j is precisely defined as $H j ≡ H j ⊗ I j ¯$⁠, where $j ¯$ denotes the complement of $H j$⁠. In the following, we will be particularly interested in situations in which the quantum systems are prepared in thermal states at inverse temperatures $β j = 1 / T j$⁠. Then, the canonical partition functions are $Z j ≡ ∑ E j e − β j E j$⁠, where ${ E j , | E j ⟩ }$ is the energy eigensystem of the Hamiltonian H_j. It will prove convenient to define the energy vectors as $E ≡ ( E 1 , … , E n ) T ∈ ℝ n$⁠, energy eigenvectors of H as $| E ⟩ ≡ | E 1 , … , E n ⟩ ∈ ℂ d$⁠, the inverse temperature vector as $β ≡ ( β 1 , … , β n ) T ∈ ℝ n$⁠, and the partition function vector as $Z ≡ ( Z 1 , … , Z n ) T ∈ ℝ n$⁠. We also define the product of the partition functions as $| Z | ≡ ∏ j = 1 n Z j$⁠.

In the present analysis, we are interested in describing the thermodynamics of the energy that is “pumped” between the subsystems $H j$⁠, and, hence, we require the total energy to be conserved, $[ U t , H ] = 0$⁠. At time $t = τ$⁠, we switch off the interaction and as always $ρ τ ≡ U τ ρ 0 eq U τ †$⁠.

Before we continue, it is important to realize that strictly speaking, Eq. (2) can be understood as the amounts of exchange thermodynamic heat only as long as the interacting V_t is essentially time-independent. In this case, V_t is suddenly turned on at t = 0 and suddenly switched off at $t = τ$⁠. In cases, for which V_t has a more complicated time-dependence, ${ Q j } j = 1 n$ is strictly speaking the changes of internal energy which may comprise heat and work. In these cases, our scenario describes a quantum energy pump. We stress that the following mathematical analysis remains identical for either case. However, to keep the conceptual arguments as accessible as possible and in close analogy to Ref. 18, we will be calling Q_j heat.

We first discuss the derivation of the quantum exchange fluctuation theorem for multipartite systems in the two-time measurement (TTM) scheme. At time t = 0, we measure the energy of the total system with the Hamiltonian H. Suppose we obtain a set of energy values $E = ( E 1 , … , E n ) T$⁠, so that the state is projected onto the eigenbases of H, $| E ⟩ ≡ | E 1 , … , E n ⟩$⁠. Then, we evolve the system under the unitary operator U_t, before we measure H again at $t = τ$⁠. The outcome of the measurement is $E ′ = ( E 1 ′ , … , E n ′ ) T$⁠, and the system's state is projected onto $| E ′ ⟩ ≡ | E 1 ′ , … , E n ′ ⟩$⁠. Then, the heat of the jth system along the trajectory is simply given by the difference in the energy measurement outcomes $E j ′ − E j$⁠. Following Ref. 18, we only focus on the weak coupling regime, where the total energy along this trajectory is approximately conserved $∑ j = 1 n ( E j ′ − E j ) ≈ 0$⁠. Note that the quantum exchange fluctuation theorem of the TTM scheme also holds for the arbitrary coupling strengths, which can be obtained by applying the characteristic function approach.⁴⁹ 

In conclusion, we have employed the one-time measurement scheme to derive the generalized quantum exchange fluctuation theorem for multipartite systems, which includes the information about the quantum correlations. From it, we have also obtained a tighter lower bound on the net entropy change, which is set by the quantum mutual information of the thermal state conditioned on the initial energy measurement outcomes. These results elucidate the role of quantum correlations in the heat exchange in a generic scenario, where the effect of the quantum correlation due to the interaction between multiple subsystems is non-negligible.

A.S. gratefully acknowledges startup funding supported by the University of Massachusetts, Boston. D.O.S.P. acknowledges the Brazilian funding agencies CNPq (Grant No. 307028/2019-4), FAPESP (Grant No. 2017/03727-0), and the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ). S.D. acknowledges support from the U.S. National Science Foundation under Grant No. DMR-2010127 and the John Templeton Foundation under Grant No. 62422.

The authors have no conflicts to disclose.

Akira Sone: Writing – original draft (lead). Diogo O. Soares-Pinto: Writing – review & editing (equal). Sebastian Deffner: Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.