Non-iid hypothesis testing: from classical to quantum
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2025
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| _version_ | 1866914079746031616 |
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| author | De Palma, Giacomo Fanizza, Marco Mowry, Connor O'Donnell, Ryan |
| author_facet | De Palma, Giacomo Fanizza, Marco Mowry, Connor O'Donnell, Ryan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-iid hypothesis testing: from classical to quantum De Palma, Giacomo Fanizza, Marco Mowry, Connor O'Donnell, Ryan Quantum Physics Data Structures and Algorithms Machine Learning 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. |
| title | Non-iid hypothesis testing: from classical to quantum |
| topic | Quantum Physics Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2510.06147 |