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arXiv:2605.31379

Rényi divergences and binary state discrimination error exponents for fermionic quasi-free states

Milán Mosonyi, Gábor Maróti-Zareczky

quant-phcs.ITmath-phSubmitted May 29, 2026• Announced Jun 1, 2026• 0 citations
v1 · May 29, 2026v2 · Sep 10, 2026
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Abstract

The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized Rényi divergences between such states, including $(α,z)$, log-Euclidean, maximal, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $α$, the regularized Rényi $(α,z)$-divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.

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38 pages

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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.
The doubly minimized Petz Renyi mutual information of order ααα is defined as the minimum of the Petz divergence of order ααα of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Renyi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Renyi mutual information of order α∈(1/2,1)α\in (1/2,1)α∈(1/2,1), and that the strong converse exponent is determined by the doubly minimized sandwiched Renyi mutual information of order α∈(1,∞)α\in (1,\infty)α∈(1,∞). This provides an operational interpretation of these types of Renyi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.
Abstract The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter α>1\alpha >1α>1 α > 1 . The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (α,z)(\alpha ,z)(α,z) ( α , z ) -divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.
The doubly minimized Petz Rényi mutual information of order α is defined as the minimum of the Petz divergence of order α of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Rényi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Rényi mutual information of order α ∈ (1/2, 1), and that the strong converse exponent is determined by the doubly minimized sandwiched Rényi mutual information of order α ∈ (1,∞). This provides an operational interpretation of these types of Rényi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.
The doubly minimized Petz Rényi mutual information of order α is defined as the minimum of the Petz divergence of order α of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Rényi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Rényi mutual information of order α ∈ (1/2, 1), and that the strong converse exponent is determined by the doubly minimized sandwiched Rényi mutual information of order α ∈ (1, ∞). This provides an operational interpretation of these types of Rényi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.
In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel Rényi$\alpha −divergencesoverall-divergences over all−divergencesoverall\alpha >1$. We prove the equality of these two threshold values (and therefore the strong converse property for this problem) using a minimax argument based on a newly established continuity property of the sandwiched Rényi divergences. Motivated by this, we give a detailed analysis of the continuity properties of various other quantum (channel) Rényi divergences, which may be of independent interest.