Super-exponential distinguishability of correlated quantum states

• URL: http://arxiv.org/abs/2203.16511v2
• Date: Mon, 23 May 2022 17:54:50 GMT
• Title: Super-exponential distinguishability of correlated quantum states
• Authors: Gergely Bunth, G\'abor Mar\'oti, Mil\'an Mosonyi, Zolt\'an Zimbor\'as
• Abstract summary: A super-exponential decrease for both types of error probabilities is only possible in the trivial case. We show that a qualitatively different behaviour can occur when there is correlation between the samples.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the problem of asymptotic binary i.i.d. state discrimination, the optimal asymptotics of the type I and the type II error probabilities is in general an exponential decrease to zero as a function of the number of samples; the set of achievable exponent pairs is characterized by the quantum Hoeffding bound theorem. A super-exponential decrease for both types of error probabilities is only possible in the trivial case when the two states are orthogonal, and hence can be perfectly distinguished using only a single copy of the system. In this paper we show that a qualitatively different behaviour can occur when there is correlation between the samples. Namely, we use gauge-invariant and translation-invariant quasi-free states on the algebra of the canonical anti-commutation relations to exhibit pairs of states on an infinite spin chain with the properties that a) all finite-size restrictions of the states have invertible density operators, and b) the type I and the type II error probabilities both decrease to zero at least with the speed $e^{-nc\log n}$ with some positive constant $c$, i.e., with a super-exponential speed in the sample size $n$. Particular examples of such states include the ground states of the $XX$ model corresponding to different transverse magnetic fields. In fact, we prove our result in the setting of binary composite hypothesis testing, and hence it can be applied to prove super-exponential distinguishability of the hypotheses that the transverse magnetic field is above a certain threshold vs. that it is below a strictly lower value.

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