Non-iid hypothesis testing: from classical to quantum

TL;DR

This work develops non-identical-source hypothesis testing for both classical distributions and quantum states. It extends known iid results by showing that, in the classical case, a tester with $c=2$ samples per source suffices when $T \gg \frac{\sqrt{d}}{\varepsilon^2}+\frac{1}{\varepsilon^4}$, while in the quantum setting a strikingly efficient regime with $c=1$ already achieves iid-optimal copy complexity, requiring $T \gg \frac{d}{\varepsilon^2}$ to distinguish $\rho_{\mathrm{avg}}$ from a target state $\sigma$ in trace distance, and similarly for $\chi^2$-divergence based tests. The paper introduces a quantum Efron--Stein inequality and a quantum Efron--Stein decomposition to bound estimation variance, enabling tight non-iid quantum testing analyses. It also extends these results to identity testing between unknown states and discusses how robustness to $\chi^2$ and Bures divergences translates into trace-distance guarantees. Collectively, the results reduce sample complexity for non-identical quantum data certification and provide a versatile toolkit for quantum statistical testing under non-iid sources.

Abstract

We study hypothesis testing (aka state certification) in the non-identically distributed setting. A recent work (Garg et al. 2023) considered the classical case, in which one is given (independent) samples from $T$ unknown probability distributions $p_1, \dots, p_T$ on $[d] = \{1, 2, \dots, d\}$, and one wishes to accept/reject the hypothesis that their average $p_{\mathrm{avg}}$ equals a known hypothesis distribution $q$. Garg et al. showed that if one has just $c = 2$ samples from each $p_i$, and provided $T \gg \frac{\sqrt{d}}{ε^2} + \frac{1}{ε^4}$, one can (whp) distinguish $p_{\mathrm{avg}} = q$ from $d_{\mathrm{TV}}(p_{\mathrm{avg}},q) > ε$. This nearly matches the optimal result for the classical iid setting (namely, $T \gg \frac{\sqrt{d}}{ε^2}$). Besides optimally improving this result (and generalizing to tolerant testing with more stringent distance measures), we study the analogous problem of hypothesis testing for non-identical quantum states. Here we uncover an unexpected phenomenon: for any $d$-dimensional hypothesis state $σ$, and given just a single copy ($c = 1$) of each state $ρ_1, \dots, ρ_T$, one can distinguish $ρ_{\mathrm{avg}} = σ$ from $D_{\mathrm{tr}}(ρ_{\mathrm{avg}},σ) > ε$ provided $T \gg d/ε^2$. (Again, we generalize to tolerant testing with more stringent distance measures.) This matches the optimal result for the iid case, which is surprising because doing this with $c = 1$ is provably impossible in the classical case. We also show that the analogous phenomenon happens for the non-iid extension of identity testing between unknown states. A technical tool we introduce may be of independent interest: an Efron-Stein inequality, and more generally an Efron-Stein decomposition, in the quantum setting.

Theorems & Definitions (48)

• theorem 1.1
• theorem 1.2
• theorem 2.1
• theorem 2.2
• theorem 2.3
• theorem 2.4
• theorem 2.5
• lemma 5.10
• proof
• definition 6.3