We establish the general framework of quantum fluctuation theorems by finding the symmetry between the forward and backward transitions of any given quantum channel. The Petz recovery map is adopted as the reverse quantum channel, and the notion of entropy production in thermodynamics is extended to the quantum regime. Our result shows that the fluctuation theorems, which are normally considered for thermodynamic processes, can be a powerful tool to study the detailed statistics of quantum systems as well as the effect of coherence transfer in an arbitrary non-equilibrium quantum process. We introduce a complex-valued entropy production to fully understand the relation between the forward and backward processes through the quantum channel. We find the physical meaning of the imaginary part of entropy production to witness the broken symmetry of the quantum channel. We also show that the imaginary part plays a crucial role in deriving the second law from the quantum fluctuation theorem. The dissipation and fluctuation of various quantum resources including quantum free energy, asymmetry and entanglement can be coherently understood in our unified framework. Our fluctuation theorem can be applied to a wide range of physical systems and dynamics to query the reversibility of a quantum state for the given quantum processing channel involving coherence and entanglement.
We establish the general framework of quantum fluctuation theorems by finding the symmetry between the forward and backward transitions of any given quantum channel. The Petz recovery map is adopted as the reverse quantum channel, and the notion of entropy production in thermodynamics is extended to the quantum regime. Our result shows that the fluctuation theorems, which are normally considered for thermodynamic processes, can be a powerful tool to study the detailed statistics of quantum systems as well as the effect of coherence transfer in an arbitrary non-equilibrium quantum process. We introduce a complex-valued entropy production to fully understand the relation between the forward and backward processes through the quantum channel. We find the physical meaning of the imaginary part of entropy production to witness the broken symmetry of the quantum channel. We also show that the imaginary part plays a crucial role in deriving the second law from the quantum fluctuation theorem. The dissipation and fluctuation of various quantum resources including quantum free energy, asymmetry and entanglement can be coherently understood in our unified framework. Our fluctuation theorem can be applied to a wide range of physical systems and dynamics to query the reversibility of a quantum state for the given quantum processing channel involving coherence and entanglement.
Entropy production and the detailed fluctuation theorem are of fundamental importance for thermodynamic processes. In this paper we study the multiple entropy production for multitime quantum processes in a unified framework. For closed quantum systems and Markovian open quantum systems, the given entropy productions all satisfy the detailed fluctuation relation. This also shows that the entropy production rate under these processes is non-negative. For non-Markovian open quantum systems, the memory effect can lead to a negative-entropy production rate. Thus, in general, the entropy production of the marginal distribution does not satisfy the detailed fluctuation theorem relation. Our framework can be applied to a wide range of physical systems and dynamics. It provides a systematic tool for studying entropy production and its rate under arbitrary quantum processes.
Abstract We introduce a fully quantum notion of entropy production based on the non-commutative extension of the classical log-ratio between forward and reverse processes. Given a pair of quantum objects associated with the forward and reverse descriptions, we define a Hermitian entropy-production operator whose expectation value is non-negative and equal to the Belavkin–Staszewski relative entropy. The operator satisfies exact integral and detailed fluctuation theorems without requiring commutativity. We then specialize this construction to the case in which the forward process is described by a single quantum channel and the reverse process is defined inferentially, through Bayesian retrodiction relative to a prior state, with the Petz transpose map as the Bayesian inverse. In this setting, the relevant quantities can be evaluated explicitly, leading to a number of natural structural and physical properties. The framework recovers the classical formula in the commutative limit, yields explicit expressions for the average entropy production, and clarifies where the fully quantum case departs from standard thermodynamic expectations.
Fluctuation theorems have elevated the second law of thermodynamics to a statistical realm by establishing a connection between time-forward and time-reversal probabilities, providing invaluable insight into nonequilibrium dynamics. While well established in classical systems, their quantum generalization, incorporating coherence and the diversity of quantum noise, remains open. We report the experimental validation of a quantum fluctuation theorem (QFT) in a photonic system, applicable to general quantum processes with nonclassical characteristics, including quasi-probabilistic descriptions of entropy production and multiple time-reversal processes. Our experiment confirms that the ratio between the quasi-probabilities of the time-forward and any multiple time-reversal processes obeys a generalized Crooks QFT. Moreover, coherence induced by a quantum process leads to the imaginary components of quantum entropy production, governing the phase factor in the QFT. These findings underscore the fundamental symmetry between a general quantum process and its time reversal, providing an elementary toolkit to explore noisy quantum information processing.
We present a general quantum fluctuation theorem for the entropy production of an open quantum system coupled to multiple environments, not necessarily at equilibrium. Such a general theorem, when restricted to the weak-coupling and Markovian regime, holds for both local and global master equations, corroborating the thermodynamic consistency of local quantum master equations. The theorem is genuinely quantum, as it can be expressed in terms of conservation of a Hermitian operator, describing the dynamics of the system state operator and of the entropy change in the baths. The integral fluctuation theorem follows from the properties of such an operator. Furthermore, it is also valid when the system is described by a time-dependent Hamiltonian. As such, the quantum Jarzynski equality is a particular case of the general result presented here. Moreover, our result can be extended to nonthermal baths, as long as microreversibility is preserved. We present some numerical examples to showcase the exact results previously obtained. We finally generalize the fluctuation theorem to the case where the interaction between the system and the bath is explicitly taken into account. We show that the fluctuation theorem amounts to a relation between time-reversed dynamics of the global density matrix and a two-time correlation function along the forward dynamics involving the baths' entropy alone.
Abstract One of the major goals of quantum thermodynamics is the characterization of irreversibility and its consequences in quantum processes. Here, we discuss how entropy production provides a quantification of the irreversibility in open quantum systems through the quantum fluctuation theorem. We start by introducing a two-time quantum measurement scheme, in which the dynamical evolution between the measurements is described by a completely positive, trace-preserving (CPTP) quantum map (forward process). By inverting the measurement scheme and applying the time-reversed version of the quantum map, we can study how this backward process differs from the forward one. When the CPTP map is unital, we show that the stochastic quantum entropy production is a function only of the probabilities to get the initial measurement outcomes in correspondence of the forward and backward processes. For bipartite open quantum systems we also prove that the mean value of the stochastic quantum entropy production is sub-additive with respect to the bipartition (except for product states). Hence, we find a method to detect correlations between the subsystems. Our main result is the proposal of an efficient protocol to determine and reconstruct the characteristic functions of the stochastic entropy production for each subsystem. This procedure enables to reconstruct even others thermodynamical quantities, such as the work distribution of the composite system and the corresponding internal energy. Efficiency and possible extensions of the protocol are also discussed. Finally, we show how our findings might be experimentally tested by exploiting the state of-the-art trapped-ion platforms.
Within quantum information, a very productive way to look at physical phenomena is under the light of the framework of resource theories. Through it, one can focus on questions such as in what sense a particular physical feature can be thought of as a resource for certain tasks, and how can it be quantified. A particularly prominent example of this is the theory of quantum entanglement. In recent years, a research program has emerged in which researchers are trying to build a resource theory relevant for thermodynamical and non-equilibrium phenomena at the quantum and nano scales. The overarching aim is to find how the perspective given by quantum information tools and techniques help us understand new and existing results of the field. In this thesis we advance these efforts by introducing different kinds of non-equilibrium fluctuations into the theory: fluctuations of work and fluctuations of states. In this way, we manage to merge a number of ideas from the field of stochastic thermodynamics into the resource-theoretic framework. We further explore the structure of this resource theory by introducing results in which the finiteness of the heat bath's size is considered, deriving corrections to well-known expressions regarding work extraction. On top of this, we also consider the usefulness within non-equilibrium thermodynamics of other tools coming from quantum information theory, such as the relative entropy and the idea of a recovery map. The connection between the recovery map and the property of quantum detailed balance (and time reversal more generally) is highlighted, and a bound on the entropy production of Davies maps is derived inspired by recent results on quantum information.