Marco Fanizza - Non-iid hypothesis testing: from classical to quantum

Wed, 05 Nov 2025 00:00:00 +0000

Non-iid hypothesis testing: from classical to quantum

This seminar, given by Marco Fanizza, will happend on 05 November 2025, at 13:0. It will take place in Room étage 1 - 25-26/105.

Find a map of the campus here.

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 $p_1, \ldots, p_T$ on $[d]={1,2, \ldots, d}$, and one wishes to accept/reject the hypothesis that their average $p_{\text {avg }}$ equals a known hypothesis distribution $q$. Garg et al. showed that if one has just $c=2$ samples from each $p_i$, and provided $T \gg \frac{\sqrt{d}}{\epsilon^2}+\frac{1}{\epsilon^4}$, one can (whp) distinguish $p_{\text {avg }}=q$ from $\mathrm{d}{\mathrm{TV}}\left(p{\text {avg }}, q\right)>\epsilon$. This nearly matches the optimal result for the classical iid setting (namely, $T \gg \frac{\sqrt{d}}{\epsilon^2}$ ). 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 $\sigma$, and given just a single copy ( $c=1$ ) of each state $\rho_1, \ldots, \rho_T$, one can distinguish $\rho_{\text {avg }}=\sigma$ from $\mathrm{D}{\mathrm{tr}}\left(\rho{\text {avg }}, \sigma\right)>\epsilon$ provided $T \gg d / \epsilon^2$. (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. Extending the iid result on identity testing between unknown states, we also show that given a single copy of each state $\rho_1, \cdots, \rho_T$ and $\sigma_1, \cdots, \sigma_T$, it is possible to distinguish between $\rho_{\text {avg }}=\sigma_{\text {avg }}$ from $\mathrm{D}{\mathrm{tr}}\left(\rho{\text {avg }}, \sigma_{\text {avg }}\right)>\epsilon$ provided $T \gg d / \epsilon^2$. 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.

Marco Fanizza | LIP6 - QI Team

Marco Fanizza - Non-iid hypothesis testing: from classical to quantum

Non-iid hypothesis testing: from classical to quantum

Abstract