Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2303.04138 · Physical Review A

One-shot and asymptotic classical capacity in general physical theories

Shintaro Minagawa, Hayato Arai

Published Jun 11, 2024• 0 citations
arXivHTMLPDF

Abstract

With the recent development of quantum information theory, some attempts have been made to construct information theory beyond quantum theory. Here, we consider hypothesis-testing relative entropy and one-shot classical capacity, that is, the optimal rate of classical information transmitted by using a single channel under the constraint of a certain error probability, in general physical theories where states and measurements are operationally defined. Then we obtain the upper bound of the one-shot classical capacity by generalizing the method given by Wang and Renner [Phys. Rev. Lett. 108, 200501 (2012)]. Also, we derive the lower bound of the capacity by showing the existence of a good code that can transmit classical information with a certain error probability. Applying the above two bounds, we prove the asymptotic equivalence between classical capacity and hypothesis-testing relative entropy in any general physical theory.

DOI
10.1103/physreva.109.062416

Related papers

Abelian Group Codes for Classical and Classical-Quantum Channels: One-shot and Asymptotic Rate Bounds

James Chin-Jen Pang, S. Sandeep Pradhan, Hessam Mahdavifar

We study the problem of transmission of information over classical and classical-quantum channels in the one-shot regime where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new input probability distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relative-entropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Furthermore, we apply the one-shot result to the asymptotic stationary memoryless setting, and establish a single-letter lower bound on group capacities for both classes of channels. Moreover, we derive a matching upper bound on the asymptotic group capacity.

arXiv:2406.13243•Jun 19, 2024•0 citations

Abelian Group Codes for Classical-Quantum Channels: One-Shot and Asymptotic Rate Bounds

James Chin-Jen Pang, S. Sandeep Pradhan, Hessam Mahdavifar

We study the problem of transmission of information over classical-quantum (CQ) channels in the one-shot regime where the underlying codes are constrained to be shiftedgroup codes. Given a groupG, a group code of lengthnis a subgroup ofGn. In the achievability part, we introduce a new collection of input probability distributions that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relative-entropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Furthermore, we apply the one-shot result to the asymptotic stationary memoryless setting, and establish a single-letter lower bound on thegroup capacityof a CQ channel. Moreover, we derive a matching upper bound on the asymptotic group capacity.

IEEE Transactions on Information Theory•Apr 23, 2026•0 citations

From asymptotic hypothesis testing to entropy inequalities

Christoph Hirche

This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi-$2$ entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.

arXiv:1812.05142•Dec 12, 2018•3 citations

Abelian Group Codes for Classical and CQ Channel Coding: One-Shot and Asymptotic Rate Bounds

James Chin-Jen Pang, S. Sandeep Pradhan, Hessam Mahdavifar

We study the one-shot channel coding problem over classical and classical-quantum channels, where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance in terms of hypothesis testing relative-entropies. In the converse part, we es-tablish bounds by leveraging a hypothesis testing-based approach. Further we apply the one-shot result to the asymptotic use case and establish the group capacities for both channels.

IEEE International Symposium on Information Theory (ISIT)•Jul 7, 2024•1 citation

Tight any-shot quantum decoupling

Mario Berta, Hao-Chung Cheng, Yongsheng Yao

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.

quant-pharXiv:2602.17430•Feb 20, 2026•0 citations

One-shot dynamical resource theory

Xiao Yuan, Pei Zeng, Minbo Gao +1 more

A fundamental problem in resource theory is to study the manipulation of the resource. Focusing on a general dynamical resource theory of quantum channels, here we consider tasks of one-shot resource distillation and dilution with a single copy of the resource. For any target of unitary channel or pure state preparation channel, we establish a universal strategy to determine upper and lower bounds on rates that convert between any given resource and the target. We show that the rates are related to resource measures based on the channel robustness and the channel hypothesis testing entropy, with regularization factors of the target resource measures. The strategy becomes optimal with converged bounds when the channel robustness is finite and measures of the target resource collapse to the same value. The single-shot result also applies to asymptotic parallel manipulation of channels to obtain asymptotic resource conversion rates. We give several examples of dynamical resources, including the purity, classical capacity, quantum capacity, non-uniformity, coherence, and entanglement of quantum channels. Our results are applicable to general dynamical resource theories with potential applications in quantum communication, fault-tolerant quantum computing, and quantum thermodynamics.

arXiv:2012.02781•Dec 4, 2020•4 citations
With the recent development of quantum information theory, some attempts have been made to construct information theory beyond quantum theory. Here, we consider hypothesis-testing relative entropy and one-shot classical capacity, that is, the optimal rate of classical information transmitted by using a single channel under the constraint of a certain error probability, in general physical theories where states and measurements are operationally defined. Then we obtain the upper bound of the one-shot classical capacity by generalizing the method given by Wang and Renner [Phys. Rev. Lett. 108, 200501 (2012)]. Also, we derive the lower bound of the capacity by showing the existence of a good code that can transmit classical information with a certain error probability. Applying the above two bounds, we prove the asymptotic equivalence between classical capacity and hypothesis-testing relative entropy in any general physical theory.
We study the problem of transmission of information over classical and classical-quantum channels in the one-shot regime where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new input probability distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relative-entropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Furthermore, we apply the one-shot result to the asymptotic stationary memoryless setting, and establish a single-letter lower bound on group capacities for both classes of channels. Moreover, we derive a matching upper bound on the asymptotic group capacity.
We study the problem of transmission of information over classical-quantum (CQ) channels in the one-shot regime where the underlying codes are constrained to be shiftedgroup codes. Given a groupG, a group code of lengthnis a subgroup ofGn. In the achievability part, we introduce a new collection of input probability distributions that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relative-entropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Furthermore, we apply the one-shot result to the asymptotic stationary memoryless setting, and establish a single-letter lower bound on thegroup capacityof a CQ channel. Moreover, we derive a matching upper bound on the asymptotic group capacity.
This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi-222 entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.
We study the one-shot channel coding problem over classical and classical-quantum channels, where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance in terms of hypothesis testing relative-entropies. In the converse part, we es-tablish bounds by leveraging a hypothesis testing-based approach. Further we apply the one-shot result to the asymptotic use case and establish the group capacities for both channels.
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.
A fundamental problem in resource theory is to study the manipulation of the resource. Focusing on a general dynamical resource theory of quantum channels, here we consider tasks of one-shot resource distillation and dilution with a single copy of the resource. For any target of unitary channel or pure state preparation channel, we establish a universal strategy to determine upper and lower bounds on rates that convert between any given resource and the target. We show that the rates are related to resource measures based on the channel robustness and the channel hypothesis testing entropy, with regularization factors of the target resource measures. The strategy becomes optimal with converged bounds when the channel robustness is finite and measures of the target resource collapse to the same value. The single-shot result also applies to asymptotic parallel manipulation of channels to obtain asymptotic resource conversion rates. We give several examples of dynamical resources, including the purity, classical capacity, quantum capacity, non-uniformity, coherence, and entanglement of quantum channels. Our results are applicable to general dynamical resource theories with potential applications in quantum communication, fault-tolerant quantum computing, and quantum thermodynamics.