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AVS Quantum Science

Fluctuation theorem for nonunital dynamics

John Goold, Kavan Kishore Modi

Published Dec 1, 2021• 10 citations
Publisher

Abstract

We derive an expression for the fluctuations of the energy in an open quantum system undergoing a nontrivial evolution. The statistics of energy are constructed by means of a two-point measurement process when the system has been evolved by a completely positive and trace-preserving map. The statistics depend only on the state of the system and the dynamical map generating the evolution. In particular, we consider the Tasaki–Crooks fluctuation theorem, where a unitary evolution is replaced by a quantum channel. Our construction of a backward process using an ancilla sheds important light on the mathematical structure at play when nonunital channels are considered in quantum thermodynamics.

DOI
10.1116/5.0065123

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We derive an expression for the fluctuations of the energy in an open quantum system undergoing a nontrivial evolution. The statistics of energy are constructed by means of a two-point measurement process when the system has been evolved by a completely positive and trace-preserving map. The statistics depend only on the state of the system and the dynamical map generating the evolution. In particular, we consider the Tasaki–Crooks fluctuation theorem, where a unitary evolution is replaced by a quantum channel. Our construction of a backward process using an ancilla sheds important light on the mathematical structure at play when nonunital channels are considered in quantum thermodynamics.
Relations similar to work and exchange fluctuations have been recently derived for open systems dynamically evolving in the presence of an ancilla. Extending these relations and constructing a non-equilibrium Helmholtz equation we derive a general expression for the energetic and entropic changes of an open quantum system undergoing a nontrivial evolution. The expressions depend only on the state of the system and the dynamical map generating the evolution. Furthermore our formalism makes no assumption on either the nature or dimension of the ancilla. Our results are expected to find application in understanding the energetics of complex quantum systems undergoing open dynamics.
In this paper, we extend the fluctuation theorems used for quantum channels to multitime processes. The fluctuation theorems for quantum channels are less restrictive. We show that the given entropy production can be equal to the result of a closed system environment. The assumption that the system evolves under a completely positive and trace-preserving map is quite general, but it is more specific for cases in which the system is initially correlated with the environment. System-environment correlations arise naturally in multitime processes with which we can give clear and physical interpretations regarding the effects of correlations. Multitime processes can provide many-body channels. The Choi state of such a many-body channel is called a process tensor. One can derive channels by executing the process tensor on a set of operations. We establish a general quantum fluctuation theorem framework for a many-body channel and its derived channels. In this framework, the effects of correlations are reflected in a Markovian property. For Markovian processes, we can extend the two-point measurement to a three-point measurement and obtain that the fluctuation theorems contain complete information about the intermediate state. For non-Markovian processes, the complete measurement of the intermediate state leads to conflicts. Therefore, we use a general measurement, which only provides partial information, for the intermediate state. The corresponding fluctuation theorems show that memory effects can reduce these fluctuations. This is consistent with the fact that system states can be recovered under non-Markovian processes.
We present fluctuation theorems and moment generating function equalities for generalized thermodynamic observables and quantum dynamics described by completely positive trace preserving maps, with and without feedback control. Our results include the quantum Jarzynski equality and Crooks fluctuation theorem, and clarify the special role played by the thermodynamic work and thermal equilibrium states in previous studies. We show that for a specific class of generalized measurements, which include projective measurements, unitality replaces microreversibility as the condition for the physicality of the reverse process in our fluctuation theorems. We present an experimental application of our theory to the problem of extracting the system-bath coupling magnitude, which we do for a system of pairs of coupled superconducting flux qubits undergoing quantum annealing.
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.
Jarzynski equality and related fluctuation theorems can be formulated for various setups. Such an equality was recently derived for nonunitary quantum evolutions described by unital quantum operations, i.e., for completely positive, trace-preserving maps, which preserve the maximally mixed state. We analyze here a more general case of arbitrary quantum operations on finite systems and derive the corresponding form of the Jarzynski equality. It contains a correction term due to nonunitality of the quantum map. Bounds for the relative size of this correction term are established and they are applied for exemplary systems subjected to quantum channels acting on a finite-dimensional Hilbert space.
A general tool for the description of open quantum systems is given by the formalism of quantum operations. The most important of these are trace-preserving maps, also known as quantum channels. We discuss those conditions on quantum channels under which the Jarzynski equality and related fluctuation theorems hold. It is essential that the representing quantum channel be unital. Under this condition, we first derive the corresponding Jarzynski equality. For a bistochastic map and its adjoint, we further formulate a theorem of Tasaki–Crooks type. In the context of unital channels, some notes on heat transfer between two quantum systems are given. We also consider the case of a finite system operated on by an external agent with feedback control. When unital channels are applied at the first stage and, for a mutual-information form, at the further ones, we obtain quantum Jarzynski–Sagawa–Ueda relations. These are extensions of the previously given results to unital quantum operations.