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:1512.02096

On the structutre of the algebra generated by the non-commutative operator graph demonstrating superactivation for a zero-error capacity

Grigori Gennadievich Amosov, Il'ya Yur'evich Zhdanovskiy

Published Dec 7, 2015• 0 citations
arXivHTMLPDF

Abstract

Recently M.E. Shirokov introduced the non-commutative operator graph depending on the complex parameter $θ$ to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We study the algebraic structure of this graph. The relations for the algebra generated by the graph are derived. In the limiting case $θ=\pm1$ the graph becomes abelian and degenerates into the direct sum of four one-dimensional irreducible representations of the Klein group.

DOI
10.48550/arxiv.1512.02096

Related papers

Structure of the algebra generated by a noncommutative operator graph which demonstrates the superactivation phenomenon for zero-error capacity

Grigori Gennadievich Amosov, Il'ya Yur'evich Zhdanovskiy

Mathematical Notes•May 1, 2016•5 citations

Sufficient conditions for additivity of the zero-error classical capacity of quantum channels

Jeonghoon Park, Jeong San Kim

The one-shot zero-error classical capacity of a quantum channel is the maximum amount of classical information that can be transmitted with zero probability of error via a single channel use. This capacity is fundamentally characterized by the logarithm of the independence number of the noncommutative graph induced by the quantum channel. Consequently, the additivity of the one-shot zero-error classical capacity is equivalent to the multiplicativity of the independence number of the associated noncommutative graphs. As the independence number is not multiplicative in general, the specific conditions under which multiplicativity is preserved remain an open question in quantum information theory. Here, we establish some sufficient conditions for the multiplicativity of the independence number and provide explicit examples of quantum channels that satisfy these criteria. Furthermore, we investigate the block form of noncommutative graphs and derive the conditions under which the independence number remains multiplicative within this framework. These results offer new insights into the structural properties of noncommutative graphs and the fundamental limits of zero-error quantum communication.

quant-pharXiv:2601.18538•Physical Review A•Jan 27, 2026•0 citations

An Extreme form of Superactivation for Quantum Zero-Error Capacities

Toby S. Cubitt, Graeme Smith

The zero-error capacity of a channel is the rate at which it can send information perfectly, with zero probability of error, and has long been studied in classical information theory. We show that the zero-error capacity of quantum channels exhibits an extreme form of non-additivity, one which is not possible for classical channels, or even for the usual capacities of quantum channels. By combining probabilistic arguments with algebraic geometry, we prove that there exist channels E1 and E2 with no zero-error classical capacity whatsoever, C_0(E1) = C_0(E2) = 0, but whose joint zero-error quantum capacity is positive, Q_0(E1 x E2) >= 1. This striking effect is an extreme from of the superactivation phenomenon, as it implies that both the classical and quantum zero-error capacities of these channels can be superactivated simultaneously, whilst being a strictly stronger property of capacities. Superactivation of the quantum zero-error capacity was not previously known.

arXiv:0912.2737•Dec 14, 2009•1 citation

On superactivation of one-shot quantum zero-error capacity and the related property of quantum measurements

Максим Евгеньевич Широков, Tatiana Shulman

We give a detailed description of a low-dimensional quantum channel (input dimension 4, Choi rank 3) demonstrating the symmetric form of superactivation of one-shot quantum zero-error capacity. This property means appearance of a noiseless (perfectly reversible) subchannel in the tensor square of a channel having no noiseless subchannels. Then we describe a quantum channel with an arbitrary given level of symmetric superactivation (including the infinite value). We also show that superactivation of one-shot quantum zero-error capacity of a channel can be reformulated in terms of quantum measurement theory as appearance of an indistinguishable subspace for the tensor product of two observables having no indistinguishable subspaces.

Problems of Information Transmission•Jul 1, 2014•7 citations

Quantum Zero-Error Source-Channel Coding and Non-Commutative Graph Theory

Dan Stahlke

Alice and Bob receive a bipartite state (possibly entangled) from some finite collection or from some subspace. Alice sends a message to Bob through a noisy quantum channel such that Bob may determine the initial state, with zero chance of error. This framework encompasses, for example, teleportation, dense coding, entanglement assisted quantum channel capacity, and one-way communication complexity of function evaluation. With classical sources and channels, this problem can be analyzed using graph homomorphisms. We show this quantum version can be analyzed using homomorphisms on non-commutative graphs (an operator space generalization of graphs). Previously the Lovász ϑ number has been generalized to non-commutative graphs; we show this to be a homomorphism monotone, thus providing bounds on quantum source-channel coding. We generalize the Schrijver and Szegedy numbers, and show these to be monotones as well. As an application, we construct a quantum channel whose entanglement assisted zero-error one-shot capacity can only be unlocked using a non-maximally entangled state. These homomorphisms allow definition of a chromatic number for non-commutative graphs. Many open questions are presented regarding the possibility of a more fully developed theory.

arXiv:1405.5254•IEEE Transactions on Information Theory•Nov 2, 2015•42 citations

An Extreme Form of Superactivation for Quantum Zero-Error Capacities

Toby S. Cubitt, Graeme Smith

The zero-error capacity of a channel is the rate at which it can send information perfectly, with zero probability of error, and has long been studied in classical information theory. We show that the zero-error capacity of quantum channels exhibits an extreme form of nonadditivity, one which is not possible for classical channels, or even for the usual capacities of quantum channels. By combining probabilistic arguments with algebraic geometry, we prove that there exist channels and with no zero-error classical capacity whatsoever, , but whose joint zero-error quantum capacity is positive, . This striking effect is an extreme form of the superactivation phenomenon, as it implies that both the classical and quantum zero-error capacities of these channels can be superactivated simultaneously, while being a strictly stronger property of capacities. Superactivation of the quantum zero-error capacity was not previously known.

IEEE Transactions on Information Theory•Feb 28, 2012•28 citations
Recently M.E. Shirokov introduced the non-commutative operator graph depending on the complex parameter θθθ to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We study the algebraic structure of this graph. The relations for the algebra generated by the graph are derived. In the limiting case θ=±1θ=\pm1θ=±1 the graph becomes abelian and degenerates into the direct sum of four one-dimensional irreducible representations of the Klein group.
The one-shot zero-error classical capacity of a quantum channel is the maximum amount of classical information that can be transmitted with zero probability of error via a single channel use. This capacity is fundamentally characterized by the logarithm of the independence number of the noncommutative graph induced by the quantum channel. Consequently, the additivity of the one-shot zero-error classical capacity is equivalent to the multiplicativity of the independence number of the associated noncommutative graphs. As the independence number is not multiplicative in general, the specific conditions under which multiplicativity is preserved remain an open question in quantum information theory. Here, we establish some sufficient conditions for the multiplicativity of the independence number and provide explicit examples of quantum channels that satisfy these criteria. Furthermore, we investigate the block form of noncommutative graphs and derive the conditions under which the independence number remains multiplicative within this framework. These results offer new insights into the structural properties of noncommutative graphs and the fundamental limits of zero-error quantum communication.
The zero-error capacity of a channel is the rate at which it can send information perfectly, with zero probability of error, and has long been studied in classical information theory. We show that the zero-error capacity of quantum channels exhibits an extreme form of non-additivity, one which is not possible for classical channels, or even for the usual capacities of quantum channels. By combining probabilistic arguments with algebraic geometry, we prove that there exist channels E1 and E2 with no zero-error classical capacity whatsoever, C_0(E1) = C_0(E2) = 0, but whose joint zero-error quantum capacity is positive, Q_0(E1 x E2) >= 1. This striking effect is an extreme from of the superactivation phenomenon, as it implies that both the classical and quantum zero-error capacities of these channels can be superactivated simultaneously, whilst being a strictly stronger property of capacities. Superactivation of the quantum zero-error capacity was not previously known.
We give a detailed description of a low-dimensional quantum channel (input dimension 4, Choi rank 3) demonstrating the symmetric form of superactivation of one-shot quantum zero-error capacity. This property means appearance of a noiseless (perfectly reversible) subchannel in the tensor square of a channel having no noiseless subchannels. Then we describe a quantum channel with an arbitrary given level of symmetric superactivation (including the infinite value). We also show that superactivation of one-shot quantum zero-error capacity of a channel can be reformulated in terms of quantum measurement theory as appearance of an indistinguishable subspace for the tensor product of two observables having no indistinguishable subspaces.
Alice and Bob receive a bipartite state (possibly entangled) from some finite collection or from some subspace. Alice sends a message to Bob through a noisy quantum channel such that Bob may determine the initial state, with zero chance of error. This framework encompasses, for example, teleportation, dense coding, entanglement assisted quantum channel capacity, and one-way communication complexity of function evaluation. With classical sources and channels, this problem can be analyzed using graph homomorphisms. We show this quantum version can be analyzed using homomorphisms on non-commutative graphs (an operator space generalization of graphs). Previously the Lovász ϑ number has been generalized to non-commutative graphs; we show this to be a homomorphism monotone, thus providing bounds on quantum source-channel coding. We generalize the Schrijver and Szegedy numbers, and show these to be monotones as well. As an application, we construct a quantum channel whose entanglement assisted zero-error one-shot capacity can only be unlocked using a non-maximally entangled state. These homomorphisms allow definition of a chromatic number for non-commutative graphs. Many open questions are presented regarding the possibility of a more fully developed theory.
The zero-error capacity of a channel is the rate at which it can send information perfectly, with zero probability of error, and has long been studied in classical information theory. We show that the zero-error capacity of quantum channels exhibits an extreme form of nonadditivity, one which is not possible for classical channels, or even for the usual capacities of quantum channels. By combining probabilistic arguments with algebraic geometry, we prove that there exist channels and with no zero-error classical capacity whatsoever, , but whose joint zero-error quantum capacity is positive, . This striking effect is an extreme form of the superactivation phenomenon, as it implies that both the classical and quantum zero-error capacities of these channels can be superactivated simultaneously, while being a strictly stronger property of capacities. Superactivation of the quantum zero-error capacity was not previously known.