Non-asymptotic assisted distillation of quantum coherence

TL;DR

This paper provides a detailed analysis of environment-assisted quantum coherence distillation in finite-copy scenarios, offering exact and bound results, and demonstrating the equivalence of free operation sets in one-shot tasks.

Contribution

It introduces a semidefinite programming bound for one-shot assisted coherence distillation, proves its tightness in low dimensions, and fully characterizes asymptotic zero-error distillation for qubits and qutrits.

Findings

Exact evaluation of one-shot assisted distillable coherence

Semidefinite programming bound tight in dimensions 2 and 3

Additivity of coherence of assistance in dimension 3

Abstract

We characterize the operational task of environment-assisted distillation of quantum coherence under different sets of free operations when only a finite supply of copies of a given state is available. We first evaluate the one-shot assisted distillable coherence exactly, and introduce a semidefinite programming bound on it in terms of a smooth entropic quantity. We prove the bound to be tight for all systems in dimensions 2 and 3, which allows us to obtain computable expressions for the one-shot rate of distillation, establish an analytical expression for the best achievable fidelity of assisted distillation for any finite number of copies, and fully solve the problem of asymptotic zero-error assisted distillation for qubit and qutrit systems. Our characterization shows that all relevant sets of free operations in the resource theory of coherence have exactly the same power in the task of one-shot assisted coherence distillation, and furthermore resolves a conjecture regarding the additivity of coherence of assistance in dimension 3.