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Anais de XXXV Simpósio Brasileiro de Telecomunicações e Processamento de Sinais

Zero-Error Capacity and Invariant Subspace of a Quantum Channel

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

Published Jan 1, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.14209/sbrt.2024.1571034730

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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.
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.
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.
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.
We begin with a detailed description of a low dimensional quantum channel (dA=4,dE=3d_A=4, d_E=3dA​=4,dE​=3) demonstrating the symmetric form of superactivation of one-shot zero-error quantum capacity. This 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 ΦΦΦ such that  Qˉ0(Φ)=0\,\bar{Q}_0(Φ)=0Qˉ​0​(Φ)=0 and  Qˉ0(Φ⊗Φ)≥log⁡n \,\bar{Q}_0(Φ\otimesΦ)\geq\log n\,Qˉ​0​(Φ⊗Φ)≥logn for any  n≤+∞\,n\leq+\inftyn≤+∞. We also show that the superactivation of one-shot zero-error quantum capacity of a channel can be reformulated in terms of quantum measurements theory as appearance of an indistinguishable subspace for tensor product of two observables having no indistinguishable subspaces.
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.