Abstract
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.
Highlights
Since Shannon’s adaptation of “entropy” as the measure of information [1], we have witnessed surprising usefulness of the mathematical descriptions of thermodynamics in developing information theory [2]
III, we demonstrate how coherences in a quantum state affect the fluctuation relation deviating from conventional fluctuation theorems (FTs) and construct the quantum FT (QFT) of entropy production based on it
IV, we introduce the complex-valued quantum entropy production in order to fully describe the transition between off-diagonal elements through the quantum channel and discuss how the QFT should be modified
Summary
Since Shannon’s adaptation of “entropy” as the measure of information [1], we have witnessed surprising usefulness of the mathematical descriptions of thermodynamics in developing information theory [2]. We establish the fluctuation relation for a linear quantum channel that reproduces the FTs in the classical thermodynamic limit. This can be done by adopting the reverse quantum process, known as the Petz recovery map [56], and generalizing the concept of entropy production to take into account coherences in a quantum system. We investigate the effect of coherences in a quantum state that makes the fluctuation relation, given by the ratio between the forward and backward transition probabilities, different from the conventional FTs. When the quantum channel induces coherence transfers, the transition between diagonal and off-diagonal elements in the density matrix of a quantum state can be understood via complex-valued quantum entropy production.
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