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arXiv:2610.16267

Classical and Quantum Information Measures Characterizing Convertibility: An Axiomatic Approach

Frits Verhagen

quant-phcs.ITmath-phmath.PRSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all $α$-$z$ relative entropies for $α<1$. This demonstrates the first operational interpretation of these relative entropies.

Comments
PhD thesis at the Centre for Quantum Technologies, National University of Singapore; contains corrections of some minor errors appearing in the published papers that this thesis is based on

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In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all ααα-zzz relative entropies for α<1α<1α<1. This demonstrates the first operational interpretation of these relative entropies.
Chapter 6 builds upon the foundation of divergences from Chapter 5, advancing into entropies and relative entropies with an axiomatic approach, and the inclusion of the additivity axiom. The chapter delves into the classical and quantum relative entropies, establishing their core properties and revealing the significance of the KL-divergence introduced in Chapter 5, notably characterized by asymptotic continuity. Quantum relative entropies are addressed as generalizations of classical ones, with a focus on the conditions necessary for these measures in the quantum framework. Several variants of relative entropies are discussed, including Renyi relative entropies and their extensions to quantum domain such as the Petz quantum Renyi divergence, minimal quantum Renyi divergence, and the maximal quantum Renyi divergence. This discourse underlines the relevance of continuity and its relation to faithfulness in relative entropies. The concept of entropy is portrayed as a measure with a broad spectrum of interpretations and applications across fields, from thermodynamics and information theory to cosmology and economics.
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.
Maximal Rényi Relative Entropy for α>2α>2α>2
Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for α∈[0,1/2)α\in[0,1/2)α∈[0,1/2) and the sandwiched Rényi relative entropy for α≥1/2α\geq 1/2α≥1/2. In contrast, the maximal extension had previously been identified only for α∈[0,2]α\in[0,2]α∈[0,2], where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for α>2α>2α>2, the maximal extension is given by the ααα-zzz Rényi relative entropy with z=α−1z=α-1z=α−1. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.
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.
One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Muet al, we show that if certain monotones—namely multivariate extensions of Rényi divergences—are strictly ordered between the two tuples, then for sufficiently largen, there exists a stochastic matrix taking then-fold Kronecker power of each input distribution to then-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.
A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.