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Physica Scripta

Quantifying noncovariance of quantum channels with respect to groups

Zijian Zhang, Linshuai Zhang, Nan Li, Shunlong Luo

Published Sep 12, 2024• 2 citations
Publisher

Abstract

Abstract A quantum channel is covariant with respect to a group if it commutes with the action of the group. In general, a quantum channel may not be covariant with respect to a given group. The degree of noncovariance can vary between different channels, and it is desirable to have a quantitative characterization for the degree of channel noncovariance. In this work, we propose a measure based on the Hilbert-Schmidt norm to quantify noncovariance of quantum channels with respect to a group and demonstrate that it satisfies several desirable properties. Compared with the existing measures of channel noncovariance, our measure applies to not only compact Lie groups but also finite groups, and it is easy to evaluate. Using this measure and its modified version together with two existing measures, we evaluate and analyze channel noncovariance through an example, finding that these measures of channel noncovariance are closely related but differ from each other. They capture different perspectives of noncovariance of quantum channels. As applications, we provide a relation between channel noncovariance and approximate quantum error correction using our measures of channel noncovariance.

DOI
10.1088/1402-4896/ad7a2b

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Abstract A quantum channel is covariant with respect to a group if it commutes with the action of the group. In general, a quantum channel may not be covariant with respect to a given group. The degree of noncovariance can vary between different channels, and it is desirable to have a quantitative characterization for the degree of channel noncovariance. In this work, we propose a measure based on the Hilbert-Schmidt norm to quantify noncovariance of quantum channels with respect to a group and demonstrate that it satisfies several desirable properties. Compared with the existing measures of channel noncovariance, our measure applies to not only compact Lie groups but also finite groups, and it is easy to evaluate. Using this measure and its modified version together with two existing measures, we evaluate and analyze channel noncovariance through an example, finding that these measures of channel noncovariance are closely related but differ from each other. They capture different perspectives of noncovariance of quantum channels. As applications, we provide a relation between channel noncovariance and approximate quantum error correction using our measures of channel noncovariance.
To perform reliable quantum computation, quantum error correction is indispensable. In certain cases, continuous covariance symmetry of the physical system can make exact error correction impossible. In this work we study the approximate error correction and covariance symmetry from the information-theoretic perspective. For general encoding and noise channels, we define a quantity named infidelity to characterize the performance of the approximate quantum error correction and quantify the noncovariance of an encoding channel with respect to a general Lie group from the asymmetry measure of the corresponding Choi state. In particular, when the encoding channel is isometric, we derive a trade-off relation between infidelity and noncovariance. Furthermore, we calculate the average infidelity and noncovariance measure for a type of random code.
Nonclassical correlations are significant physical resources with extensive applications in quantum information processing. We introduce the modified Wigner-Yanase-Dyson skew information of a quantum state relative to a quantum channel, and a quantitative measure of quantum correlations. Their basic properties are explored in detail. Through a specific example, we also compare our correlations measure with the existing one. Moreover, the correlations relative to various channels including the von Neumann measurements, the unitary channels and the twirling channels are analyzed.
Abstract We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional C∗C^*C∗ C ∗ -algebras) carry an action of a compact quantum group G , and all channels (completely positive maps preserving a certain G -invariant functional) are covariant with respect to the G -actions. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint. Some key results are the following: (1) We give a necessary and sufficient condition for a covariant quantum relation to be the underlying relation of a covariant channel. (2) We show that every quantum confusability graph with a G -action (which we call a quantum G -graph) arises as the confusability graph of a covariant channel. (3) We show that a covariant channel is reversible precisely when its confusability G -graph is discrete. (4) When G is quasitriangular (this includes all compact groups), we show that covariant zero-error source-channel coding schemes are classified by covariant homomorphisms between confusability G -graphs.
In contrast to unitary evolutions, which are reversible, generic quantum processes (operations and quantum channels) are often irreversible. However, the degree of irreversibility is different for different channels, and it is desirable to have a quantitative characterization of irreversibility. In this paper, by exploiting the channel–state duality implemented by the Jamiołkowski–Choi isomorphism, we quantify the irreversibility of channels via entropy of the Jamiołkowski–Choi states of the corresponding channels and compare it with the notions of entanglement fidelity and entropy exchange. General properties of a reasonable measure of irreversibility are discussed from an intuitive perspective, and entropic measures of irreversibility are introduced. Several relations between irreversibility, entanglement fidelity, the degree of nonunitality, and decorrelating power are established. Some measures of irreversibility for a variety of prototypical channels are evaluated explicitly, revealing some information-theoretic aspects of the structure of channels from the perspective of irreversibility.
Quantum channels, in general, will disturb quantum states and will lead to decoherence (i.e., information leakage to the environment). For bipartite systems, local channels often cause more decoherence for global states than for local states due to correlations. Intuitively, the difference between these two kinds of decoherence (i.e., global and local ones) may be regarded as a quantifier characterizing certain aspects of correlations in the global bipartite states. Based on this idea, we introduce a quantifier of correlations (relative to a local channel) as the coherence difference and investigate its basic properties. We probe and quantify correlations relative to various channels including the unitary channels, the twirling channels, the projective measurements, and the weak measurements. In particular, we show that both product states and some natural classical-quantum states can be operationally characterized in terms of local channels: The product states are just those states with vanishing coherence difference relative to the twirling channel induced by the unitary group, whereas these natural classical-quantum states are just those states with vanishing coherence difference relative to the twirling channel induced by the subgroup of the unitary group that does not disturb the local states. We further illustrate the results by various examples.
We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation (U) of a finite group (G), whenever U⊗Uc is simply reducible (with Uc being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of any such linear map. The eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group G. This in turn yields necessary and sufficient conditions on the eigenvalues of any such linear map for it to be a quantum channel. We also obtain a wide class of quantum channels which are irreducibly covariant by construction. For two-dimensional irrreducible representations of the symmetric group S(3), and the quaternion group Q, we also characterize quantum channels which are both irreducibly covariant and entanglement breaking.