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arXiv:1206.3056 · Open Physics

Notes on entropic characteristics of quantum channels

Alexey E. Rastegin

Published Oct 24, 2012• 4 citations
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Abstract

Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.

DOI
10.2478/s11534-012-0134-8

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Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.
In classical information theory, the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov–Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes–Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy were proposed [10,[Formula: see text]3,[Formula: see text]8,[Formula: see text]39,[Formula: see text]15,[Formula: see text]44,[Formula: see text]9,[Formula: see text]27,[Formula: see text]28,[Formula: see text]2,[Formula: see text]19,[Formula: see text]45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S-mixing entropy by Ohya [26,[Formula: see text]27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S-mixing entropy. In this paper, based on research into the dynamical entropy and mean mutual entropy of quantum systems, this study investigates the characteristics of quantum channels that exhibit entanglement, a property unique to quantum systems, and investigates the behaviour of quantum entropy for compound quantum channels by changing the perspective from information transmission between input and output to a new perspective of information transmission from the initial state to the final state.
We consider properties of quantum channels with the use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map ( q , s )-entropy is naturally defined as the unified ( q , s )-entropy of a rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. The known inequality of Lindblad with the entropy exchange is generalized to many of the unified entropies. For the tensor product of a pair of quantum channels, we derive a two-sided estimate on the output entropy of a maximally entangled input state.
Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamio{\l}kowski-Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix obtained as reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver (q,s)(q,s)(q,s)-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.
Quantitative analysis of discontinuity of information characteristics of quantum states and channels is presented. Estimates for discontinuity jump (loss) of the von Neumann entropy for a given converging sequence of states are obtained. It is shown, in particular, that for any sequence the loss of entropy is upper bounded by the loss of mean energy (with the coefficient characterizing the Hamiltonian of a system). Then we prove that discontinuity jumps of basic measures of classical and quantum correlations in composite quantum systems are upper bounded by the loss of one of the marginal entropies (with a corresponding coefficient). Quantitative discontinuity analysis of the output entropy of a quantum operation and of basic information characteristics of a quantum channel considered as functions of a pair (channel, input state) is presented.
Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamiołkowski–Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix, obtained as the reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver ( q , s )-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.