Continuous thermomajorization and a complete set of laws for Markovian thermal processes

Abstract

The standard dynamical approach to quantum thermodynamics is based on Markovian master equations describing the thermalization of a system weakly coupled to a large environment, and on tools such as entropy production relations. Here we introduce a new framework overcoming the limitations that the current dynamical and information theory approaches encounter when applied to this setting. More precisely, based on a newly introduced notion of continuous thermomajorization, we obtain necessary and sufficient conditions for the existence of a Markovian thermal process transforming between given initial and final energy distributions of the system. These lead to a complete set of generalized entropy production inequalities including the standard one as a special case. Importantly, these conditions can be reduced to a finitely verifiable set of constraints governing non-equilibrium transformations under master equations. What is more, the framework is also constructive, i.e., it returns explicit protocols realizing any allowed transformation. These protocols use as building blocks elementary thermalizations, which we prove to be universal controls. Finally, we also present an algorithm constructing the full set of energy distributions achievable from a given initial state via Markovian thermal processes and provide a Mathematica implementation solving d = 6 on a laptop computer in minutes.

Non-Markovian thermal operations boosting the performance of quantum heat engines

“…Let us start our discussion with a rigorous definition of the distinction between Markovian and non-Markovian state transformations. The thermal operation is deemed to be Markovian when it can be expressed as the time ordered exponential [77]…”

“…Here and from hereon = 1 is taken. The Lindblad generator is required to fulfill two properties [77]:…”

“…As shown by Lostaglio and Korzekwa [77], any Markovian thermal operation on a n-level quantum system can be realized by a sequence of full or partial thermalizations acting on pairs of energy levels. Thus, in particular, a Markovian thermal operation acting an a two-level system (qubit) corresponds to its full or partial thermalization.…”