Fluctuation Theorems for Quantum Maps

Gonzalo Manzano, Jordan M. Horowitz, Juan M. R. Parrondo

We derive a general fluctuation theorem for quantum maps. The theorem applies to a broad class of quantum dynamics, such as unitary evolution, decoherence, thermalization, and other types of evolution for quantum open systems. The theorem reproduces well-known fluctuation theorems in a single and simplified framework and extends the Hatano-Sasa theorem to quantum nonequilibrium processes. Moreover, it helps to elucidate the physical nature of the environment that induces a given dynamics in an open quantum system.

Ken Funo, Masahito Ueda, Takahiro Sagawa

Zhiqiang Huang

The detailed fluctuation theorem implies symmetry in the generating function of entropy production probability. The integral fluctuation theorem directly follows from this symmetry and the normalization of the probability. In this paper, we rewrite the generating function by integrating measurements and evolution into a constructed mapping. This mapping is completely positive, and the original integral fluctuation theorem is determined by the trace-preserving property of these constructed maps. We illustrate the convenience of this method by discussing the eigenstate fluctuation theorem and heat exchange between two baths. This set of methods is also applicable to the generating functions of quasiprobability, where we observe the Petz recovery map arising naturally from this approach.

Martí Perarnau-Llobet

John Goold, Kavan Kishore Modi

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.

Mirko Degli Esposti, S. Graffi