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arXiv:2107.08036 · Communications in Mathematical Physics

The Strong Converse Exponent of Discriminating Infinite-Dimensional Quantum States

Milán Mosonyi

Published Mar 10, 2023• 11 citations
arXivHTMLPDF

Abstract

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 $$\alpha >1$$ 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 $$(\alpha ,z)$$ ( α , z ) -divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.

DOI
10.1007/s00220-022-04598-1

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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.
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.
We prove that the regularized, stabilized sandwiched Rényi divergence of finite-dimensional quantum channels converges to their regularized relative entropy as the Rényi order tends to one. The key tool is the channel hockey-stick divergence: Gour's Stinespring approximation bound and a Schatten norm estimate amplify an asymptotic bound below one into exponential decay at higher threshold rates. For channel pairs with finite max-relative entropy, known operational connections then give exponential strong converses for parallel and adaptive discrimination, a sharp zero--one testing law, and the subchannel asymptotic equipartition property.
We extend from the hyperfinite setting to general von Neumann algebras Mosonyi and Ogawa's (2015) and Mosonyi and Hiai's (2023) results showing the operational interpretation of sandwiched relative Rényi entropy in the strong converse of hypothesis testing. The specific task is to distinguish between two quantum states given many copies. We use a reduction method of Haagerup, Junge, and Xu (2010) to approximate relative entropy inequalities in an arbitrary von Neumann algebra by those in finite von Neumann algebras. Within these finite von Neumann algebras, it is possible to approximate densities via finite spectrum operators, after which the quantum method of types reduces them to effectively commuting subalgebras. Generalizing beyond the hyperfinite setting shows that the operational meaning of sandwiched Rényi entropy is not restricted to the matrices but is a more fundamental property of quantum information. Furthermore, applicability in general von Neumann algebras opens potential new connections to random matrix theory and the quantum information theory of fundamental physics.
We prove an integral representation of the ααα-zzz Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray z=cαz=cαz=cα, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders ααα above one, we require the divergence to be finite, whereas for orders below one support inclusion suffices. An explicit boundary correction removes the lower-order support assumption. The proof uses a compatible weighted Haagerup LpL^pLp scale and applies outside the data-processing region. For faithful density matrices, we express the derivative of the escort relative entropy with respect to the Rényi order in terms of a nonnegative fluctuation quantity. Differences of order-one derivatives of the divergence give quantum Fisher information and Wigner-Yanase skew information. For faithful bipartite density matrices, we also express optimized conditional ααα-zzz entropies at orders above one as averages of ordinary conditional entropies. Further applications concern coherent-state divergences in quantum field theory and the optimizing order for the strong converse exponent in hypothesis-testing.
We introduce a new quantum Rényi divergence Dα# for α∈(1,∞) defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched α -Rényi divergence between quantum channels for α>1 . Second it allows us to prove a chain rule property for the sandwiched α -Rényi divergence for α>1 which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.
We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders α∈[12,∞)α\in[\frac{1}{2},\infty)α∈[21​,∞). For α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1), the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for α∈[1,∞)α\in[1,\infty)α∈[1,∞), it is characterized by the order-ααα sandwiched R{é}nyi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched R{é}nyi divergence for α∈(2,∞)α\in(2,\infty)α∈(2,∞), expressed in terms of the corresponding order-ααα sandwiched R{é}nyi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the KKK-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1).