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:0706.1815 · Physical Review A

Entanglement measures and approximate quantum error correction

Francesco Buscemi

Published Jan 9, 2008• 14 citations
arXivHTMLPDF

Abstract

It is shown that if the loss of entanglement along a quantum channel is sufficiently small, then approximate quantum error correction is possible, thereby generalizing what happens for coherent information. Explicit bounds are obtained for the entanglement of formation and the distillable entanglement, and their validity naturally extends to other bipartite entanglement measures in between. Robustness of derived criteria is analyzed and their tightness compared. Finally, as a by-product, we prove a bound quantifying how large the gap between the entanglement of formation and distillable entanglement can be for any given finite dimensional bipartite system, thus providing a sufficient condition for distillability in term of the entanglement of formation.

DOI
10.1103/physreva.77.012309

Related papers

Entanglement measures and quantum channels

Michael D. Westmoreland, Benjamin W. Schumacher

Like the coherent information, the entanglement of formation may be used as a measure of the quantum information sent over a channel. These measures are related, and share the property of giving necessary and sufficient conditions for perfect quantum error correction.

Optical Fiber Communication Conference•Jan 1, 2001•0 citations

Approximate quantum error correction, random codes, and quantum channel capacity

Rochus Klesse

We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding attains information rates of the regularized coherent information.

arXiv:quant-ph/0701102•Physical Review A•Jun 13, 2007•39 citations

Degradable states and one-way entanglement distillation

Felix Leditzky, Nilanjana Datta, Graeme Smith

We derive an upper bound on the one-way distillable entanglement of bipartite quantum states. To this end, we revisit the notion of degradable, conjugate degradable, and antidegrad-able bipartite quantum states [1]. We prove that for degradable and conjugate degradable states the one-way distillable entanglement is equal to the coherent information, and thus given by a single-letter formula. Furthermore, it is well-known that the one-way distillable entanglement of antidegradable states is zero. We use these results to derive an upper bound for arbitrary bipartite quantum states, which is based on a convex decomposition of a bipartite state into degradable and antidegradable states. This upper bound is always at least as good an upper bound as the entanglement of formation. Applying our bound to the qubit depolarizing channel, we obtain an upper bound on its quantum capacity that is strictly better than previously known bounds in the high noise regime. We also transfer the concept of approximate degradability [2] to quantum states and show that this yields another easily computable upper bound on the one-way distillable entanglement. Moreover, both methods of obtaining upper bounds on the one-way distillable entanglement can be combined into a generalized one.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2017•2 citations

Entanglement theory and its applications in Gaussian quantum information

Spyros Tserkis

Entanglement is a physical property, emerging from the superposition of composite quantum systems, and manifested through non-classical correlations of quantum observables. In the field of Quantum Information, entanglement is considered the resource for various quantum protocols, so the ability to quantify it would allow us to associate it with the success of those protocols. Entanglement of formation is a proper way to quantify entanglement, but an analytical expression for this measure exists only for special cases. In this thesis, we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for this measure that get tight for several special cases. We further study how quantum teleportation and entanglement distillation can be employed in order to error-correct information encoded on a Gaussian state that suffers from Gaussian noise. In particular, we derive every physical state able to simulate a given phase-insensitive Gaussian channel through teleportation with finite-energy resources, and show how error-correction of a state is related to the simulation of a less decohering channel that the state has to pass through. We also discuss how the premise that the whole environment is under control of the adversary in quantum key distribution can be eliminated if we consider a teleportation-based eavesdropping attack. More specifically, we propose an all-optical teleportation attack that under collective measurements can reach optimality in the limit of infinite amount of entanglement, while for finite entanglement resources it outperforms the corresponding optimal individual attack. Finally, we show how using a class of finite-energy resource states we can increasingly approximate the infinite-energy bounds for decreasing purity, so that they provide tight upper bounds to the secret-key capacity of single-mode phase-insensitive Gaussian channels.

•Mar 27, 2020•0 citations

Unextendible Entanglement of Quantum Channels

Vishal Singh, Mark M. Wilde

Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.

arXiv:2407.15944•IEEE Transactions on Information Theory•May 7, 2025•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
It is shown that if the loss of entanglement along a quantum channel is sufficiently small, then approximate quantum error correction is possible, thereby generalizing what happens for coherent information. Explicit bounds are obtained for the entanglement of formation and the distillable entanglement, and their validity naturally extends to other bipartite entanglement measures in between. Robustness of derived criteria is analyzed and their tightness compared. Finally, as a by-product, we prove a bound quantifying how large the gap between the entanglement of formation and distillable entanglement can be for any given finite dimensional bipartite system, thus providing a sufficient condition for distillability in term of the entanglement of formation.
We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding attains information rates of the regularized coherent information.
We derive an upper bound on the one-way distillable entanglement of bipartite quantum states. To this end, we revisit the notion of degradable, conjugate degradable, and antidegrad-able bipartite quantum states [1]. We prove that for degradable and conjugate degradable states the one-way distillable entanglement is equal to the coherent information, and thus given by a single-letter formula. Furthermore, it is well-known that the one-way distillable entanglement of antidegradable states is zero. We use these results to derive an upper bound for arbitrary bipartite quantum states, which is based on a convex decomposition of a bipartite state into degradable and antidegradable states. This upper bound is always at least as good an upper bound as the entanglement of formation. Applying our bound to the qubit depolarizing channel, we obtain an upper bound on its quantum capacity that is strictly better than previously known bounds in the high noise regime. We also transfer the concept of approximate degradability [2] to quantum states and show that this yields another easily computable upper bound on the one-way distillable entanglement. Moreover, both methods of obtaining upper bounds on the one-way distillable entanglement can be combined into a generalized one.
Entanglement is a physical property, emerging from the superposition of composite quantum systems, and manifested through non-classical correlations of quantum observables. In the field of Quantum Information, entanglement is considered the resource for various quantum protocols, so the ability to quantify it would allow us to associate it with the success of those protocols. Entanglement of formation is a proper way to quantify entanglement, but an analytical expression for this measure exists only for special cases. In this thesis, we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for this measure that get tight for several special cases. We further study how quantum teleportation and entanglement distillation can be employed in order to error-correct information encoded on a Gaussian state that suffers from Gaussian noise. In particular, we derive every physical state able to simulate a given phase-insensitive Gaussian channel through teleportation with finite-energy resources, and show how error-correction of a state is related to the simulation of a less decohering channel that the state has to pass through. We also discuss how the premise that the whole environment is under control of the adversary in quantum key distribution can be eliminated if we consider a teleportation-based eavesdropping attack. More specifically, we propose an all-optical teleportation attack that under collective measurements can reach optimality in the limit of infinite amount of entanglement, while for finite entanglement resources it outperforms the corresponding optimal individual attack. Finally, we show how using a class of finite-energy resource states we can increasingly approximate the infinite-energy bounds for decreasing purity, so that they provide tight upper bounds to the secret-key capacity of single-mode phase-insensitive Gaussian channels.
Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.
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.