Lower bounds on the error probability of multiple quantum channel discrimination by the Bures angle and the trace distance

TL;DR

This paper derives lower bounds on the error probability in quantum channel discrimination using Bures angle and trace distance, proving optimality of Grover's search for fixed marked elements and outperforming recent bounds in numerical tests.

Contribution

It introduces new lower bounds for quantum channel discrimination error probabilities based on geometric metrics, extending previous results and demonstrating their effectiveness.

Findings

Lower bounds derived from Bures angle and trace distance.

Proof of Grover's search optimality for fixed marked elements.

Numerical results showing bounds outperform recent methods.

Abstract

Quantum channel discrimination is a fundamental problem in quantum information science. In this study, we consider general quantum channel discrimination problems, and derive the lower bounds of the error probability. Our lower bounds are based on the triangle inequalities of the Bures angle and the trace distance. As a consequence of the lower bound based on the Bures angle, we prove the optimality of Grover's search if the number of marked elements is fixed to some integer $\ell$. This result generalizes Zalka's result for $\ell=1$. We also present several numerical results in which our lower bounds based on the trace distance outperform recently obtained lower bounds.