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arXiv:2611.02425

Finite attainment and efficient computation of regularized quantum channel relative entropy

Yiwei Xu, Kun Fang

quant-phcs.ITSubmitted Nov 2, 2026• Announced Nov 3, 2026

Abstract

Quantum channel relative entropy can be nonadditive, making its regularization essential for asymptotic analysis. We study when the regularized value is attained at finite blocklength and how efficiently it can be approximated. For the first question, we establish a sharp dichotomy: the regularized value is either attained already at the single-copy level, implying additivity at every blocklength, or is never attained at any finite blocklength. Notably, a simple two-copy additivity test distinguishes between these alternatives for qubit channels and for channel pairs in arbitrary dimensions admitting an interior optimizer. We further show that both the interiority condition and finite nonattainment are generic, holding for almost every channel pair. For the second question, we establish explicit convergence rates for approximating the regularized channel relative entropy. For fixed channel dimensions, $\widetilde O(\varepsilon^{-3/2})$ channel copies suffice to approximate the regularized value to additive accuracy $\varepsilon$. Alternatively, a semidefinite hierarchy achieves the same accuracy at level $\widetilde O(\varepsilon^{-2})$, yielding a certified polynomial-time approximation. Together, these results reveal the intriguing structure and generic behavior of regularized channel relative entropy, sharpening our understanding of both its analytical and computational aspects.

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Quantum channel relative entropy can be nonadditive, making its regularization essential for asymptotic analysis. We study when the regularized value is attained at finite blocklength and how efficiently it can be approximated. For the first question, we establish a sharp dichotomy: the regularized value is either attained already at the single-copy level, implying additivity at every blocklength, or is never attained at any finite blocklength. Notably, a simple two-copy additivity test distinguishes between these alternatives for qubit channels and for channel pairs in arbitrary dimensions admitting an interior optimizer. We further show that both the interiority condition and finite nonattainment are generic, holding for almost every channel pair. For the second question, we establish explicit convergence rates for approximating the regularized channel relative entropy. For fixed channel dimensions,

\widetilde O(\varepsilon^{-3/2})

channel copies suffice to approximate the regularized value to additive accuracy

\varepsilon

. Alternatively, a semidefinite hierarchy achieves the same accuracy at level

\widetilde O(\varepsilon^{-2})

, yielding a certified polynomial-time approximation. Together, these results reveal the intriguing structure and generic behavior of regularized channel relative entropy, sharpening our understanding of both its analytical and computational aspects.

The quantum relative entropy is a fundamental quantity in quantum information science, characterizing the distinguishability between two quantum states. However, this quantity is not additive in general for correlated quantum states, necessitating regularization for precise characterization of the operational tasks of interest. Recently, we proposed the study of the regularized relative entropy between two sequences of sets of quantum states in [arXiv:2411.04035], which captures a general framework for a wide range of quantum information tasks. Here, we show that given suitable structural assumptions and efficient descriptions of the sets, the regularized relative entropy can be efficiently approximated within an additive error by a quantum relative entropy program of polynomial size. This applies in particular to the regularized relative entropy in adversarial quantum channel discrimination. Moreover, we apply the idea of efficient approximation to quantum resource theories. In particular, when the set of interest does not directly satisfy the required structural assumptions, it can be relaxed to one that does. This provides improved and efficient bounds for the entanglement cost of quantum states and channels, entanglement distillation and magic state distillation. Numerical results demonstrate improvements even for the first level of approximation.

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.

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of

ε

in

\mathrm{poly}(1/ε, \text{input dimension})

time.

The study of entanglement inevitably leads to the concept of regularization, where measures are evaluated on infinitely many copies of a state. Single-letter formulas try to make these asymptotic entanglement quantities accessible through a calculation on one copy of a state, but are rarely available. We introduce a decreasing hierarchy of computable upper bounds on the asymptotic relative entropy of entanglement with respect to positive-partial-transpose (PPT) states. The regularization of every fixed hierarchy level equals this asymptotic quantity. Additivity at any level therefore yields a single-letter formula, even when the usual one-copy relative entropy is nonadditive. We establish such additivity at the first nontrivial level for two broad multiparameter families, both containing all Werner states. The upper-bound construction also extends to sandwiched Renyi divergences. The hierarchy motivates

k

-multinegative states, a generalization of binegative states and the associated

k

-multinegativity, which gives explicit upper bounds on both the asymptotic relative entropy and exact PPT entanglement cost. We construct states which are

k

-multinegative at arbitrarily large depths

k

that separate consecutive levels of the previously introduced entanglement-cost hierarchy, disproving its conjectured finite collapse. These results provide state-dependent single-letter formulas while identifying a limitation of universal finite-level characterizations.

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of$\epsilon

in

\mathrm {poly}(1/\epsilon, \text {input dimension})

time, compared to the

\exp (1/\epsilon, \text {input dimension})$running time required for direct computation.

We show that the regularized channel Rényi divergence is continuous at

α= 1

. As consequences, we establish an exponentially strong converse for channel discrimination, a subchannel-smoothed asymptotic equipartition property, a relative entropy accumulation theorem, and a new proof of the generalized quantum Stein's lemma. Our result further shows that the regularized channel relative entropy is computable. The continuity proof relies on a variational formula that expresses the regularized channel divergence as a single-letter optimization over variables of unrestricted finite dimension. We then establish continuity using a novel spectral confinement technique.