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arXiv:2610.06842

Quantum information processing under maximal distance measures

Bartosz Regula, Ludovico Lami, Mario Berta, Marco Tomamichel

quant-phmath-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

The measures of distance frequently used in quantum information, such as trace distance or fidelity-based purified distance, are the smallest extensions of the corresponding distance measures in classical information theory. However, this way of extending classical distances is not unique, and a whole spectrum of other extensions can be defined. Here we study the question of how quantum information processing tasks are affected by a change of the error metrics to the maximal extensions of the classical distance measures, imposing more demanding error requirements on operational tasks. We find that the performance of some tasks, such as channel coding or quantum resource distillation, is not affected whatsoever by this change. However, for other tasks such as channel simulation and quantum resource dilution, we show that the rates are now given not by functions of the standard (Umegaki) quantum relative entropy, but ones based on the Belavkin-Staszewski relative entropy. Our results connect the Belavkin-Staszewski relative entropy with operational tasks and reveal that known reversibility results in quantum information, including the reverse Shannon theorem and the asymptotic reversibility of quantum resource transformations, no longer hold under maximal distance measures.

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The measures of distance frequently used in quantum information, such as trace distance or fidelity-based purified distance, are the smallest extensions of the corresponding distance measures in classical information theory. However, this way of extending classical distances is not unique, and a whole spectrum of other extensions can be defined. Here we study the question of how quantum information processing tasks are affected by a change of the error metrics to the maximal extensions of the classical distance measures, imposing more demanding error requirements on operational tasks. We find that the performance of some tasks, such as channel coding or quantum resource distillation, is not affected whatsoever by this change. However, for other tasks such as channel simulation and quantum resource dilution, we show that the rates are now given not by functions of the standard (Umegaki) quantum relative entropy, but ones based on the Belavkin-Staszewski relative entropy. Our results connect the Belavkin-Staszewski relative entropy with operational tasks and reveal that known reversibility results in quantum information, including the reverse Shannon theorem and the asymptotic reversibility of quantum resource transformations, no longer hold under maximal distance measures.
We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (“generalized transition probability”) of Uhlmann, is topologically equivalent to the trace-norm distance.
Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.
We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e., an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a technique to extend pure-state entanglement measures to mixed bipartite states.
In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard [Formula: see text]-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal [Formula: see text]-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.
Chapter 5 delves into divergences and distance measures, which are crucial for comparing quantum states. It begins with classical divergences such as the Kullback–Leibler and Jensen–Shannon, then advances to their quantum counterparts, discussing their optimal characteristics. Influenced by quantum resource theories, these quantum extensions provide foundational insights into the robust tools of resource theories. The chapter concentrates on particular divergences that serve as true metrics, including the trace distance and a variant of the fidelity, and explores the concept of distance between subnormalized states, which is essential in the context of quantum measurements. It emphasizes the purified distance, a useful tool for understanding the entanglement cost of quantum systems, setting the stage for further exploration in later chapters. The chapter offers a mathematically approachable survey of these measures, underscoring their practical importance in quantum information theory.
An elementary algebraic approach to unified quantum information theory is given. The operational meaning of entanglement as specifically quantum encoding is disclosed. General relative entropy as information divergence is introduced, and three most important types of relative information, namely, the Araki-Umegaki type (A-type), the Belavkin-Staszewski type (B-type), and the thermodynamical (C-type) are discussed. It is shown that true quantum entanglement-assisted entropy is greater than semiclassical (von Neumann) quantum entropy, and the proper positive quantum conditional entropy is introduced. The general quantum mutual information via entanglement is defined, and the corresponding types of quantum channel capacities as a supremum via the generalized encodings are formulated. The additivity problem for quantum logarithmic capacities for products of arbitrary quantum channels under appropriate constraints on encodings is discussed. It is proved that true quantum capacity, which is achieved on the standard entanglement as an optimal quantum encoding, retains the additivity property of the logarithmic quantum channel entanglement-assisted capacities on the products of quantum input states. This result for quantum logarithmic information of A-type, which was obtained earlier by the author, is extended to any type of quantum information.