Simple upper and lower bounds on the ultimate success probability for discriminating arbitrary finite-dimensional quantum processes

TL;DR

This paper derives simple upper and lower bounds on the maximum success probability for discriminating finite-dimensional quantum processes, including complex strategies, with bounds that are computationally efficient and broadly applicable.

Contribution

It introduces novel, simple bounds on quantum process discrimination success probabilities, applicable to arbitrary processes and adaptable to general operational theories.

Findings

Upper bounds are simple and tight, independent of quantum phenomena.

Lower bounds based on Bayesian updating are computationally efficient.

Bounds are applicable to processes with internal memories and multiple time steps.

Abstract

We consider the problem of discriminating finite-dimensional quantum processes, also called quantum supermaps, that can consist of multiple time steps. Obtaining the ultimate performance for discriminating quantum processes is of fundamental importance, but is challenging mainly due to the necessity of considering all discrimination strategies allowed by quantum mechanics, including entanglement-assisted strategies and adaptive strategies. In the case in which the processes to be discriminated have internal memories, the ultimate performance would generally be more difficult to analyze. In this paper, we present a simple upper bound on the ultimate success probability for discriminating arbitrary quantum processes. In the special case of multi-shot channel discrimination, it can be shown that the ultimate success probability increases by at most a constant factor determined by the given channels if the number of channel evaluations increases by one. We also present a lower bound based on Bayesian updating, which has a low computational cost. Our numerical experiments demonstrate that the proposed bounds are reasonably tight. The proposed bounds do not explicitly depend on any quantum phenomena, and can be readily extended to a general operational probabilistic theory.