Pretty Good Bounds on the worst-case Pretty Good Measurement

TL;DR

The paper addresses worst-case quantum state discrimination among $m$ pure states and derives a tighter lower bound on the Pretty Good Measurement (PGM) success probability than the Gram-matrix bound for $m \ge 4$, expressed via $F = \max_{i\neq j} |\langle v_i|v_j\rangle|^2$. The authors adapt the Barnum-Knill average-case analysis to the worst-case setting and compare PGM to the sequential measurement algorithm, yielding a bound $P_\mathrm{PGM} \ge \dfrac{\left(1-4(m-1)F^2\right)^2}{1+mF^2}$, which scales as $1 - O(F^2)$ for small $F$ and improves upon $1 - mF$ when $m \ge 4$. This sharpens the theoretical understanding of PGM in adversarial or noisy environments and highlights practical advantages of a fixed, single-shot POVM implementation over SMA in noisy quantum devices.

Abstract

We derive a new lower bound on the success probability of the Pretty Good Measurement (PGM) for worst-case quantum state discrimination among $m$ quantum 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.

Figures (1)

• Figure 1: Plot of PGM success probabilities as a function of $F$ for multiple $m$ values. The linear terms are given by $1 - mF$ and the refined terms are given by $\frac{\left(1-4\left(m-1\right)F^2\right)^2}{1+mF^2}$. For $m\geq 4$, we see that our refined bound always outperforms the linear bound (as proved explicitly in App. \ref{['sec:proof_of_improved_bound']}).

Theorems & Definitions (5)

• Definition 2.1: Worst-case PGM
• Definition 2.2: SMA
• Theorem 3.1
• proof
• Remark 3.1