Optimal unitary trajectories under commuting target and cost observables; applications to cooling

TL;DR

This paper derives the optimal unitary operations for quantum state transformations under commuting observables, including energy conservation, with applications to cooling in finite quantum systems.

Contribution

It provides a comprehensive solution for optimal quantum state transformations with commuting target and cost observables, extending to include a third conserved quantity.

Findings

Optimal unitaries for state transformation under commuting observables derived

Extension of results to include energy conservation constraints

Application demonstrated in ground state cooling scenarios

Abstract

The preparation of quantum states, especially cooling, is a fundamental technology for nanoscale devices. The past decade has seen important results related to both the limits of state transformation and the limits to their efficiency -- the quantum versions of the third and second law of thermodynamics. The limiting cases always involve an infinite resource cost, typically machine complexity or time. Realistic state preparation takes into account both a finite size of the machine and constraints on the operations we can perform. In this work, we determine in full generality the optimal operation for a predominant quantum paradigm: state transformation under a single unitary operation upon a finite system, in the case where the observables corresponding to the target (such as ground state probability) and cost (such as dissipation) commute. We then extend this result to the case of having a third, commuting, globally conserved quantity (such as total energy). The results are demonstrated with the paradigmatic example of ground state cooling, for both arbitrary and energy-preserving unitary operations.