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Anais de XXXV Simpósio Brasileiro de Telecomunicações e Processamento de Sinais

A condition for the zero-error capacity of quantum channels

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

Published Jan 1, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.14209/sbrt.2023.1570917602

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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, 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, 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 study zero-error communication and exact simulation of quantum channels assisted by quantum no-signalling correlations. For a channel with Kraus operator space KKK, we introduce a semidefinite programming parameter Γ(K)Γ(K)Γ(K) that is exactly multiplicative under tensor products and upper-bounds ΘNS(K)Θ_{\rm NS}(K)ΘNS​(K), where log⁡2ΘNS(K)=C0,NS(K)\log_2Θ_{\rm NS}(K)=C_{0,\rm NS}(K)log2​ΘNS​(K)=C0,NS​(K) and C0,NS(K)C_{0,\rm NS}(K)C0,NS​(K) denotes the asymptotic quantum no-signalling-assisted zero-error capacity. Among known upper bounds on ΘNSΘ_{\rm NS}ΘNS​ given by semidefinite programs, ΓΓΓ is the tightest. We also prove that the previously introduced simulation lower bound Σ−(K)Σ^{-}(K)Σ−(K) is exactly multiplicative and satisfies ΘNS(K)≤Γ(K)≤Σ−(K)≤Γ(K)2Θ_{\rm NS}(K)\leqΓ(K)\leqΣ^{-}(K)\leqΓ(K)^2ΘNS​(K)≤Γ(K)≤Σ−(K)≤Γ(K)2. Using this multiplicativity, we show that neither S0,NS(K)=log⁡2Σ−(K)S_{0,\rm NS}(K)=\log_2Σ^{-}(K)S0,NS​(K)=log2​Σ−(K) nor its regularized extension S0,NS(K)=lim⁡n→∞n−1log⁡2Σ−(K⊗n)S_{0,\rm NS}(K)=\lim\nolimits_{n\to\infty}n^{-1}\log_2Σ^{-}(K^{\otimes n})S0,NS​(K)=limn→∞​n−1log2​Σ−(K⊗n) holds in general. This answers an open question about the asymptotic simulation cost S0,NS(K)S_{0,\rm NS}(K)S0,NS​(K). For classical-quantum channels, we obtain sharp bounds. If rrr is the effective output dimension, then ΘNS≤Γ≤(1+r1/2)ΘNS/2Θ_{\rm NS}\leqΓ\leq(1+r^{1/2})Θ_{\rm NS}/2ΘNS​≤Γ≤(1+r1/2)ΘNS​/2 and Γ≤Σ−≤(1+r1/2)Γ/2Γ\leqΣ^{-}\leq(1+r^{1/2})Γ/2Γ≤Σ−≤(1+r1/2)Γ/2. If kkk is the minimum rank of the output support projections, then Σ−/ΘNS≤[(1+r1/2)/(1+k1/2)]2Σ^{-}/Θ_{\rm NS}\leq[(1+r^{1/2})/(1+k^{1/2})]^2Σ−/ΘNS​≤[(1+r1/2)/(1+k1/2)]2, and this bound is sharp for every 1≤k≤r1\leq k\leq r1≤k≤r. Thus, we obtain the sharp bound S0,NS−C0,NS≤2log⁡2[(1+r1/2)/(1+k1/2)]S_{0,\rm NS}-C_{0,\rm NS}\leq2\log_2[(1+r^{1/2})/(1+k^{1/2})]S0,NS​−C0,NS​≤2log2​[(1+r1/2)/(1+k1/2)].
The aim of this work is to study the zero-error capacity of pure-state classical-quantum channels in the setting of list decoding. We provide an achievability bound for list-size two and a converse bound holding for every fixed list size. The two bounds coincide for channels whose pairwise absolute state overlaps form a positive semi-definite matrix. Finally, we discuss a remarkable peculiarity of the classical-quantum case: differently from the fully classical setting, the rate at which the sphere-packing bound diverges might not be achievable by zero-error list codes, even when we take the limit of fixed but arbitrarily large list size.
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.