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

IEEE Access

Shemesh Theorem and Its Relation With the Zero-Error Quantum Information Theory

Marciel Medeiros de Oliveira, Micael Andrade Dias, Andresso da Silva, Francisco de Assis

Published Jan 1, 2024• 1 citation
Publisher

Abstract

In this paper, our objective is to establish results that connect Shemesh theorem with quantum zero-error information theory. To achieve this, we will first introduce definitions and properties related to the zero-error capacity of a quantum channel, with particular emphasis on demonstrating that the eigenstates common to the Kraus operators representing the quantum channel are fixed points. This result has previously been employed to develop a test for determining whether a quantum channel possesses positive zero-error capacity. In this work, we demonstrate that the eigenstates common to the Kraus operators representing the channel reside within the Shemesh subspace. Consequently, the fixed points of the quantum channel act as a bridge between Shemesh theorem and quantum zero-error information theory, thereby enabling the derivation of new results on the zero-error capacity of quantum channels.

DOI
10.1109/access.2024.3514518

Related papers

A condition for the zero-error capacity of quantum channels

Marciel Medeiros de Oliveira, Francisco de Assis, Micael Andrade Dias

In this paper, we present a condition for the zero-error capacity of quantum channels. To achieve this result we first prove that the eigenvectors (or eigenstates) common to the Kraus operators representing the quantum channel are fixed points of the channel. From this fact and assuming that these Kraus operators have at least two eigenstates in common and also considering that every quantum channel has at least one fixed point, it is proved that the zero-error capacity of the quantum channel is positive. Moreover, this zero-error capacity condition is a lower bound for the zero-error capacity of the quantum channel. This zero-error capacity condition of quantum channels has a peculiar feature that it is easy to verify when one knows the Kraus operators representing the quantum channel.

arXiv:2312.13406•Dec 20, 2023•0 citations

A condition for the zero-error capacity of quantum channels

Marciel Medeiros de Oliveira, Francisco de Assis, Micael Andrade Dias

In this paper, we present a condition for the zeroerror capacity of quantum channels.To achieve this result we first prove that the eigenvectors (or eigenstates) common to the Kraus operators representing the quantum channel are fixed points of the channel.From this fact and assuming that these Kraus operators have at least two eigenstates in common and also considering that every quantum channel has at least one fixed point, it is proved that the zero-error capacity of the quantum channel is positive.Moreover, this zero-error capacity condition is a lower bound for the zero-error capacity of the quantum channel.This zero-error capacity condition of quantum channels has a peculiar feature that it is easy to verify when one knows the Kraus operators representing the quantum channel.

Anais de XXXV Simpósio Brasileiro de Telecomunicações e Processamento de Sinais•Jan 1, 2023•1 citation

Application of Shemesh theorem to quantum channels

Michał Białończyk, Andrzej Jamiołkowski, Karol Życzkowski

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.

arXiv:1711.02354•Journal of Mathematical Physics•Oct 1, 2018•6 citations

Zero-Error Capacity and Invariant Subspace of a Quantum Channel

Marciel Medeiros de Oliveira, Andresso da Silva, Micael Andrade Dias +1 more

In this paper, we explore the connection between the zero-error capacity and the invariant subspace of a quantum channel.We begin by demonstrating that a quantum channel with two invariant subspaces, which only have trivial intersections, has a positive zero-error capacity.Additionally, we establish that an unital quantum channel with a non-trivial invariant subspace of at least two dimensions also has a positive zero-error capacity.

Anais de XXXV Simpósio Brasileiro de Telecomunicações e Processamento de Sinais•Jan 1, 2024•0 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

On the Mathematical Boundaries of Communication with Zero-Capacity Quantum Channels

László Gyöngyösi, Sándor Imre

In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered. These phenomena are purely quantum mechanical and completely unimaginable in classical systems. Recently, the most important discovery in Quantum Information Theory was the possibility of transmitting quantum information over zero-capacity quantum channels. In this work we prove that the possibility of superactivation of quantum channel capacities is determined by the mathematical properties of the quantum relative entropy function.

EPiC series in computing•Jan 23, 2018•1 citation
In this paper, our objective is to establish results that connect Shemesh theorem with quantum zero-error information theory. To achieve this, we will first introduce definitions and properties related to the zero-error capacity of a quantum channel, with particular emphasis on demonstrating that the eigenstates common to the Kraus operators representing the quantum channel are fixed points. This result has previously been employed to develop a test for determining whether a quantum channel possesses positive zero-error capacity. In this work, we demonstrate that the eigenstates common to the Kraus operators representing the channel reside within the Shemesh subspace. Consequently, the fixed points of the quantum channel act as a bridge between Shemesh theorem and quantum zero-error information theory, thereby enabling the derivation of new results on the zero-error capacity of quantum channels.
In this paper, we present a condition for the zero-error capacity of quantum channels. To achieve this result we first prove that the eigenvectors (or eigenstates) common to the Kraus operators representing the quantum channel are fixed points of the channel. From this fact and assuming that these Kraus operators have at least two eigenstates in common and also considering that every quantum channel has at least one fixed point, it is proved that the zero-error capacity of the quantum channel is positive. Moreover, this zero-error capacity condition is a lower bound for the zero-error capacity of the quantum channel. This zero-error capacity condition of quantum channels has a peculiar feature that it is easy to verify when one knows the Kraus operators representing the quantum channel.
In this paper, we present a condition for the zeroerror capacity of quantum channels.To achieve this result we first prove that the eigenvectors (or eigenstates) common to the Kraus operators representing the quantum channel are fixed points of the channel.From this fact and assuming that these Kraus operators have at least two eigenstates in common and also considering that every quantum channel has at least one fixed point, it is proved that the zero-error capacity of the quantum channel is positive.Moreover, this zero-error capacity condition is a lower bound for the zero-error capacity of the quantum channel.This zero-error capacity condition of quantum channels has a peculiar feature that it is easy to verify when one knows the Kraus operators representing the quantum channel.
Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.
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.
In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered. These phenomena are purely quantum mechanical and completely unimaginable in classical systems. Recently, the most important discovery in Quantum Information Theory was the possibility of transmitting quantum information over zero-capacity quantum channels. In this work we prove that the possibility of superactivation of quantum channel capacities is determined by the mathematical properties of the quantum relative entropy function.