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arXiv:2509.21308 · IEEE Transactions on Information Theory

Efficient Quantum Measurements: Computational Max- and Measured Rényi Divergences and Applications

Álvaro Yángüez, Thomas A. Hahn, Jan Kochanowski

Published Apr 2, 2026• 0 citations
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Abstract

Quantum information processing is limited, in practice, to efficiently implementable operations. This motivates the study of quantum divergences that preserve their operational meaning while faithfully capturing these computational constraints. Using geometric, computational, and information theoretic tools, we define two new types of computational divergences, which we termcomputational max-divergence and computational measured Rényi divergences. Both are constrained by a family of efficient binary measurements, and thus useful for state discrimination tasks in the computational setting. We prove that, in the infinite-order limit, the computational measured Rényi divergence coincides with the computational max-divergence, mirroring the corresponding relation in the unconstrained information-theoretic setting. For the many-copy regime, we introduce regularized versions and establish a one-sided computational Stein bound on achievable hypothesis-testing exponents under efficient measurements, giving the regularized computational measured relative entropy an operational meaning. We further define resource measures induced by our computational divergences and prove an asymptotic continuity bound for the computational measured relative entropy of resource. Focusing on entanglement, we relate our results to previously proposed computational entanglement measures and provide explicit separations from the information-theoretic setting. Together, these results provide a principled, cohesive approach towards state discrimination tasks and resource quantification under computational constraints.

DOI
10.1109/tit.2026.3680247

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Quantum information processing is limited, in practice, to efficiently implementable operations. This motivates the study of quantum divergences that preserve their operational meaning while faithfully capturing these computational constraints. Using geometric, computational, and information theoretic tools, we define two new types of computational divergences, which we termcomputational max-divergence and computational measured Rényi divergences. Both are constrained by a family of efficient binary measurements, and thus useful for state discrimination tasks in the computational setting. We prove that, in the infinite-order limit, the computational measured Rényi divergence coincides with the computational max-divergence, mirroring the corresponding relation in the unconstrained information-theoretic setting. For the many-copy regime, we introduce regularized versions and establish a one-sided computational Stein bound on achievable hypothesis-testing exponents under efficient measurements, giving the regularized computational measured relative entropy an operational meaning. We further define resource measures induced by our computational divergences and prove an asymptotic continuity bound for the computational measured relative entropy of resource. Focusing on entanglement, we relate our results to previously proposed computational entanglement measures and provide explicit separations from the information-theoretic setting. Together, these results provide a principled, cohesive approach towards state discrimination tasks and resource quantification under computational constraints.
Abstract Having a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties, which are not satisfied by other quantum Rényi divergences. For example we prove a chain rule inequality that immediately implies the “amortization collapse” for the geometric Rényi divergence, addressing an open question by Berta et al. [Letters in Mathematical Physics 110:2277–2336, 2020, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter and efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.
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 propose an extension of the classical Rényi divergences to quantum states through an optimization over probability distributions induced by restricted sets of measurements. In particular, we define the notion of locally-measured Rényi divergences, where the set of allowed measurements originates from variants of locality constraints between (distant) parties AAA and BBB. We then derive variational bounds on the locally-measured Rényi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured Rényi divergences on variants of highly symmetric data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements. For nnn-fold tensor powers, we further employ our variational formulae to derive corresponding additivity results, which gives the locally-measured Rényi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.
We propose an extension of the classical Rényi divergences to quantum states through an optimization over probability distributions induced by restricted sets of measurements. In particular, we define the notion of locally-measured Rényi divergences, where the set of allowed measurements originates from variants of locality constraints between (distant) partiesAandB. We then derive variational bounds on the locally-measured Rényi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured Rényi divergences on variants of highly symmetric data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements. For n-fold tensor powers, we further employ our variational formulae to derive corresponding additivity results, which gives the locally-measured Rényi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.
Quantum relative entropy is a core concept in physics, governing the limits of communication, thermodynamic irreversibility and quantum resource conversion. However, the requirement that physical processes cannot increase state distinguishability, the data-processing inequality, permits an infinite family of alternative divergence measures. Here we show that quantum relative entropy is uniquely selected by a sharper operational principle. We evaluate distinguishability through binary guessing games, in which an observer discriminates between pairs of quantum states using the optimal measurement. We prove that any additive measure that respects the odds revealed by these optimal measurements must coincide with the Umegaki relative entropy. This rigidity is a purely quantum phenomenon. Whereas classical theory permits a continuous family of valid divergence measures, including Rényi divergences, quantum noncommutativity. collapses this mathematical freedom. The result is exact, requiring neither a thermodynamic limit of infinitely many copies nor super-additivity assumptions for correlated states. It establishes quantum relative entropy not merely as an asymptotic quantity, but as the unique additive distinguishability measure compatible with single-shot quantum discrimination.
Quantum generalizations of Rényi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Rényi divergence Dˉα\bar{D}_αDˉα​ and the minimal quantum Rényi divergence D~α\widetilde{D}_αDα​. Moreover, the maximum quantum Rényi divergence D^α\widehat{D}_αDα​ is of particular mathematical interest. In this Master thesis, we investigate relations between these divergences and their applications in quantum information theory. Our main result is a reverse Araki-Lieb-Thirring inequality that implies a new relation between the minimal and the Petz divergence, namely that αDˉα(ρ∥σ)⩽D~α(ρ∥σ)α\bar{D}_α(ρ\| σ) \leqslant \widetilde{D}_α(ρ\| σ)αDˉα​(ρ∥σ)⩽Dα​(ρ∥σ) for α∈[0,1]α\in [0,1]α∈[0,1] and where ρρρ and σσσ are density operators. This bound suggests defining a "pretty good fidelity", whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction. In addition, we provide a new proof of the inequality D~1(ρ∥σ)⩽D^1(ρ∥σ) ,\widetilde{D}_{1}(ρ\| σ) \leqslant \widehat{D}_{1}(ρ\| σ)\, ,D1​(ρ∥σ)⩽D1​(ρ∥σ), based on the Araki-Lieb-Thirring inequality. This leads to an elegant proof of the logarithmic form of the reverse Golden-Thompson inequality.