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Time-reversal and strong H-theorem for quantum discrete-time Markov channels

Francesco Ticozzi, Michele Pavon

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The time reversal of a completely-positive, nonequilibrium discrete-time quantum Markov evolution is derived via a suitable adjointness relation. Space-time harmonic processes are introduced for the forward and reverse-time transition mechanisms, and their role for relative entropy dynamics is discussed.

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Time reversal and duality of Markov processes will play a key role in the stochastic theory of quantum physics. We first discuss time reversal of time-homogeneous Markov processes. To handle the time reversal and (space-time) duality of time-inhomogeneous Markov processes we will define the so-called space-time Markov processes, and prove the main theorems of time reversal and duality. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The dynamics of an open quantum system can be described by a quantum operation: A linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes toward equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.
The theory of Schrödinger bridges for diffusion processes is extended to classical and quantum Markov chains. Taking into account the past-future lack of symmetry of the discrete-time setting, results bear a striking resemblance to the classical ones. In particular, the solution of the path space maximum entropy problems is always obtained from the a priori model by means of a suitable multiplicative functional transformation. In the quantum case, nonequilibrium time reversal of quantum channels is discussed and space-time harmonic processes are introduced.
The dynamics of an open quantum system can be described by a quantum operation: A linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes toward equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.
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.