Pretty Good Bounds on the worst-case Pretty Good Measurement

TL;DR

This paper establishes a tighter lower bound on the success probability of the Pretty Good Measurement for worst-case quantum state discrimination, especially in low-fidelity regimes, improving understanding of quantum measurement limits.

Contribution

It introduces a new lower bound for the PGM's success probability that surpasses previous bounds for multiple states, using techniques from average-case analysis.

Findings

New lower bound tighter than previous for m≥4

Success probability decreases quadratically with maximum overlap in low-fidelity regime

Bound applies to worst-case quantum state discrimination

Abstract

We derive a new lower bound on the success probability of the Pretty Good Measurement (PGM) for worst-case quantum state discrimination among $m$ pure states. Our bound is strictly tighter than the previously known Gram-matrix-based bound for $m\geq 4$. The proof adapts techniques from Barnum and Knill's analysis of the average-case PGM, applied here to the worst-case scenario. By comparing the PGM to the sequential measurement algorithm, we obtain a guarantee showing that, in the low-fidelity regime, the PGM's success probability decreases quadratically with respect to the maximum pairwise overlap, rather than linearly as in earlier bounds.