Generalised quantum Sanov theorem revisited
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915538218778624 |
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| author | Lami, Ludovico |
| author_facet | Lami, Ludovico |
| contents | Given two families of quantum states $A$ and $B$, called the null and the alternative hypotheses, quantum hypothesis testing is the task of determining whether an unknown quantum state belongs to $A$ or $B$. Mistaking $A$ for $B$ is a type I error, and vice versa for the type II error. In quantum Shannon theory, a fundamental role is played by the Stein exponent, i.e. the asymptotic rate of decay of the type II error probability for a given threshold on the type I error probability. Stein exponents have been thoroughly investigated -- and, sometimes, calculated. However, most currently available solutions apply to settings where the hypotheses simple (i.e. composed of a single state), or else the families $A$ and $B$ need to satisfy stringent constraints that exclude physically important sets of states, such as separable states or stabiliser states. In this work, we establish a general formula for the Stein exponent where both hypotheses are allowed to be composite: the alternative hypothesis $B$ is assumed to be either composite i.i.d. or arbitrarily varying, with components taken from a known base set, while the null hypothesis $A$ is fully general, and required to satisfy only weak compatibility assumptions that are met in most physically relevant cases -- for instance, by the sets of separable or stabiliser states. Our result extends and subsumes the findings of [BBH, CMP 385:55, 2021] (that we also simplify), as well as the 'generalised quantum Sanov theorem' of [LBR, arXiv:2408.07067]. The proof relies on a careful quantum-to-classical reduction via measurements, followed by an application of the results on classical Stein exponents obtained in [Lami, arXiv:today]. We also devise new purely quantum techniques to analyse the resulting asymptotic expressions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalised quantum Sanov theorem revisited Lami, Ludovico Quantum Physics Information Theory Mathematical Physics Given two families of quantum states $A$ and $B$, called the null and the alternative hypotheses, quantum hypothesis testing is the task of determining whether an unknown quantum state belongs to $A$ or $B$. Mistaking $A$ for $B$ is a type I error, and vice versa for the type II error. In quantum Shannon theory, a fundamental role is played by the Stein exponent, i.e. the asymptotic rate of decay of the type II error probability for a given threshold on the type I error probability. Stein exponents have been thoroughly investigated -- and, sometimes, calculated. However, most currently available solutions apply to settings where the hypotheses simple (i.e. composed of a single state), or else the families $A$ and $B$ need to satisfy stringent constraints that exclude physically important sets of states, such as separable states or stabiliser states. In this work, we establish a general formula for the Stein exponent where both hypotheses are allowed to be composite: the alternative hypothesis $B$ is assumed to be either composite i.i.d. or arbitrarily varying, with components taken from a known base set, while the null hypothesis $A$ is fully general, and required to satisfy only weak compatibility assumptions that are met in most physically relevant cases -- for instance, by the sets of separable or stabiliser states. Our result extends and subsumes the findings of [BBH, CMP 385:55, 2021] (that we also simplify), as well as the 'generalised quantum Sanov theorem' of [LBR, arXiv:2408.07067]. The proof relies on a careful quantum-to-classical reduction via measurements, followed by an application of the results on classical Stein exponents obtained in [Lami, arXiv:today]. We also devise new purely quantum techniques to analyse the resulting asymptotic expressions. |
| title | Generalised quantum Sanov theorem revisited |
| topic | Quantum Physics Information Theory Mathematical Physics |
| url | https://arxiv.org/abs/2510.06340 |