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arXiv:1306.4261

Quantum generalized fluctuation-dissipation relations in terms of time-distributed observations

Yu. E. Kuzovlev

Published Jun 18, 2013• 0 citations
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Abstract

New formulations of quantum generalized fluctuation-dissipation relations in terms of characteristic and probabilistic functionals of continuous observations are suggested and discussed. It is shown that control of entropy production in quantum system turns any measurement in it to a source of its extra perturbations, and because of this effect relations between probabilities of mutually time-reversed processes become definitely non-local in their functional space

DOI
10.48550/arxiv.1306.4261

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New formulations of quantum generalized fluctuation-dissipation relations in terms of characteristic and probabilistic functionals of continuous observations are suggested and discussed. It is shown that control of entropy production in quantum system turns any measurement in it to a source of its extra perturbations, and because of this effect relations between probabilities of mutually time-reversed processes become definitely non-local in their functional space
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.
We propose an alternative notion of time reversal in open quantum systems as represented by linear quantum operations, and a related generalization of classical entropy production in the environment. This functional is the ratio of the probability to observe a transition between two states under the forward and the time reversed dynamics, and leads, as in the classical case, to fluctuation relations as tautological identities. As in classical dynamics in contact with a heat bath, time reversal is not unique, and we discuss several possibilities. For any bistochastic map its dual map preserves the trace and describes a legitimate dynamics reversed in time, in that case the entropy production in the environment vanishes. For a generic stochastic map we construct a simple quantum operation which can be interpreted as a time reversal. For instance, the decaying channel, which sends the excited state into the ground state with a certain probability, can be reversed into the channel transforming the ground state into the excited state with the same probability.
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.
The thermodynamic behavior of Markovian open quantum systems can be described at the level of fluctuations by using continuous monitoring approaches. However, practical applications require assessing imperfect detection schemes, where the definition of main thermodynamic quantities becomes subtle and universal fluctuation relations are unknown. Here, we fill this gap by deriving a universal fluctuation relation that links thermodynamic entropy production and information-theoretical irreversibility along single trajectories in inefficient monitoring setups. This relation provides as a corollary an irreversibility estimator of dissipation using imperfect detection records that lower bounds the underlying entropy production at the level of visible trajectories. We illustrate our findings with a driven-dissipative two-level system following quantum jump trajectories and discuss the experimental applicability of our results for thermodynamic inference.