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The strong converse rate of quantum hypothesis testing for correlated quantum states.

Milán Mosonyi, Tomohiro Ogawa

Published Jul 14, 2014• 4 citations
OpenAlex

Abstract

We present two general approaches to obtain the strong converse rate of quantum hypothesis testing for correlated quantum states. One approach requires that the states satisfy a certain factorization property; typical examples of such states are the temperature states of translation-invariant finite-range interactions on a spin chain. The other approach requires the differentiability of a regularized R\'enyi $\alpha$-divergence in the parameter $\alpha$; typical examples of such states include temperature states of non-interacting fermionic lattice systems, and classical irreducible Markov chains. In all cases, we get that the strong converse exponent is equal to the Hoeffding anti-divergence, which in turn is obtained from the regularized R\'enyi divergences of the two states.

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arXiv:2201.05477•IEEE Transactions on Information Theory•Sep 26, 2022•14 citations
We present two general approaches to obtain the strong converse rate of quantum hypothesis testing for correlated quantum states. One approach requires that the states satisfy a certain factorization property; typical examples of such states are the temperature states of translation-invariant finite-range interactions on a spin chain. The other approach requires the differentiability of a regularized R'enyi α\alphaα-divergence in the parameter α\alphaα; typical examples of such states include temperature states of non-interacting fermionic lattice systems, and classical irreducible Markov chains. In all cases, we get that the strong converse exponent is equal to the Hoeffding anti-divergence, which in turn is obtained from the regularized R'enyi divergences of the two states.
We present two general approaches to obtain the strong converse exponent of simple quantum hypothesis testing for correlated quantum states. One approach requires that the states satisfy a certain factorization property; typical examples of such states are the temperature states of translation-invariant finite-range interactions on a spin chain. The other approach requires the differentiability of a regularized Rényi α-divergence in the parameter α; typical examples of such states include temperature states of non-interacting fermionic lattice systems, and classical irreducible Markov chains. In all cases, we get that the strong converse exponent is equal to the Hoeffding antidivergence, which in turn is obtained from the regularized Rényi divergences of the two states.
We study the error exponents in quantum hypothesis testing between two sets of quantum states, extending the analysis beyond the independent and identically distributed case to encompass composite correlated hypotheses. In particular, we introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states, establishing their equivalence or quantitative relations. In the error exponent regime, we generalize the quantum Hoeffding bound to stable sequences of convex, compact sets of quantum states, demonstrating that the optimal Type-I error exponent, under an exponential constraint on the Type-II error, is precisely characterized by the regularized quantum Hoeffding divergence between the sets. In the strong converse exponent regime, we establish a general lower bound on the exponent in terms of the regularized quantum Hoeffding anti-divergence, and we prove a matching upper bound when the null hypothesis is a singleton, under additional assumptions. The generality of these results enables applications in various contexts, including (i) refining the generalized quantum Stein’s lemma by [Fang, Fawzi & Fawzi, 2024]; (ii) exhibiting counterexamples to the continuity of the regularized Petz R´enyi divergence and Hoeffding divergence; (iii) obtaining error exponents for adversarial channel discrimination and resource detection problems.
Martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning. Here, we apply techniques borrowed from this field to quantum hypothesis testing, which is the problem of discriminating quantum states belonging to two different sequences, (ρn}n and (σn}n. We obtain achievability bounds on the finite blocklength type II Steinand Hoeffding errors which, for i.i.d. states, are in general tighter than the corresponding bounds obtained by Audenaert, Mosonyi, and Verstraete. We also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a (non-homogeneous) factorization property. Examples of such sequences include Gibbs states of spin chains with translation-invariant finite range interaction, as well as finitely correlated quantum states. We apply our results to find bounds on the capacity of a certain class of classical-quantum channels with memory, which satisfy a so-called channel factorization property-both in the finite blocklength and moderate deviation regimes.
One possibility of defining a quantum Rényi$\alpha −divergenceoftwoquantumstatesistooptimizetheclassicalReˊnyi-divergence of two quantum states is to optimize the classical Rényi−divergenceoftwoquantumstatesistooptimizetheclassicalReˊnyi\alpha −divergenceoftheirpost−measurementprobabilitydistributionsoverallpossiblemeasurements(measuredReˊnyidivergence),andmayberegularizethesequantitiesovermultiplecopiesofthetwostates(regularizedmeasuredReˊnyi-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi−divergenceoftheirpost−measurementprobabilitydistributionsoverallpossiblemeasurements(measuredReˊnyidivergence),andmayberegularizethesequantitiesovermultiplecopiesofthetwostates(regularizedmeasuredReˊnyi\alpha −divergence).AkeyobservationbehindthetheoremforthestrongconverseexponentofasymptoticbinaryquantumstatediscriminationisthattheregularizedmeasuredReˊnyi-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi−divergence).AkeyobservationbehindthetheoremforthestrongconverseexponentofasymptoticbinaryquantumstatediscriminationisthattheregularizedmeasuredReˊnyi\alpha −divergencecoincideswiththesandwichedReˊnyi-divergence coincides with the sandwiched Rényi−divergencecoincideswiththesandwichedReˊnyi\alpha −divergencewhen-divergence when−divergencewhen\alpha >1.Moreover,italsofollowsfromthesametheoremthattoachievethis,itissufficienttoconsider2−outcomemeasurements(tests)foranynumberofcopies(thisissomewhatsurprising,asachievingthemeasuredReˊnyi. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi.Moreover,italsofollowsfromthesametheoremthattoachievethis,itissufficienttoconsider2−outcomemeasurements(tests)foranynumberofcopies(thisissomewhatsurprising,asachievingthemeasuredReˊnyi\alpha −divergencefor-divergence for−divergenceforncopiesmightrequireanumberofmeasurementoutcomesthatdivergesincopies might require a number of measurement outcomes that diverges incopiesmightrequireanumberofmeasurementoutcomesthatdivergesinn,ingeneral).Inviewofthis,itseemsnaturaltoexpectthesamewhen, in general). In view of this, it seems natural to expect the same when,ingeneral).Inviewofthis,itseemsnaturaltoexpectthesamewhen\alpha < 1;however,weshowthatthisisnotthecase.Infact,weshowthatevenforcommutingstates(classicalcase)theregularizedquantityattainableusing2−outcomemeasurementsisingeneralstrictlysmallerthantheReˊnyi; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi;however,weshowthatthisisnotthecase.Infact,weshowthatevenforcommutingstates(classicalcase)theregularizedquantityattainableusing2−outcomemeasurementsisingeneralstrictlysmallerthantheReˊnyi\alpha −divergence(whichisuniqueintheclassicalcase).Inthegeneralquantumcasethisshowsthattheabove“regularizedtest−measured”Reˊnyi-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi−divergence(whichisuniqueintheclassicalcase).Inthegeneralquantumcasethisshowsthattheabove“regularizedtest−measured”Reˊnyi\alpha −divergenceisnotevenaquantumextensionoftheclassicalReˊnyidivergencewhen-divergence is not even a quantum extension of the classical Rényi divergence when−divergenceisnotevenaquantumextensionoftheclassicalReˊnyidivergencewhen\alpha < 1,insharpcontrasttothe, in sharp contrast to the,insharpcontrasttothe\alpha >1$case.