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arXiv:1302.5336

On equalities in two entropic inequalities

Максим Евгеньевич Широков

Published Feb 21, 2013• 1 citation
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Abstract

A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained. It implies that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion of existence of such states is given. All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy.

DOI
10.48550/arxiv.1302.5336

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A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained. It implies that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion of existence of such states is given. All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy.
We obtain a simple criterion for local equality between the con- strained Holevo capacity and the quantum mutual information of a quan- tum channel. This shows that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion for the existence of such states is given. All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy. Bibliography: 20 titles.
It is easy to show coincidence of the entanglement-assisted classical capacity and the Holevo capacity for any c-q channel between finite dimensional quantum systems. In this paper we prove the converse assertion: coincidence of the above-mentioned capacities of a quantum channel implies that the χχχ-essential part of this channel is a c-q channel (the χχχ-essential part is a restriction of a channel obtained by discarding all input states useless for transmission of classical information). The above observations are generalized to infinite dimensional quantum channels with linear constraints by using the obtained conditions for coincidence of the quantum mutual information and the constrained Holevo capacity.
The bosonic quantum entropy power inequality bounds the entropy of a beam-splitter output in terms of the input entropies. We determine its complete equality class among independent inputs of finite mean photon number: for any number of modes and any transmissivity strictly between zero and one, equality in either the linear or the exponential form holds if and only if the inputs are Gaussian with the same covariance matrix, allowing arbitrary displacements. The main step shows that equality for one output forces the two outputs to be independent: a thermal auxiliary converts equality into preservation of mutual information by a fixed noisy channel, and a two-outcome instrument shows that preservation of mutual information requires a product state with the reference. The quantum Darmois--Skitovich theorem then gives Gaussianity, while a central-limit argument reduces exponential equality to linear equality.
Several relations between the Holevo capacity and the entanglement-assisted classical capacity of a quantum channel are proved, necessary and sufficient conditions for their coincidence are obtained. In particular, it is shown that these capacities coincide if (correspondingly, only if) the channel (correspondingly, the χχχ--essential part of the channel) belongs to the class of classical-quantum channels (the χχχ-essential part is a restriction of a channel obtained by discarding all states useless for transmission of classical information). The obtained conditions and their corollaries are generalized to channels with linear constraints. By using these conditions it is shown that the question of coincidence of the Holevo capacity and the entanglement-assisted classical capacity depends on the constraint (even for classical-quantum channels). Properties of the difference between the quantum mutual information and the χχχ-function (constrained Holevo capacity) of a quantum channel are explored.
Information transmission via a quantum channel is one of the most important processes considered by the quantum information theory. Quantum channels obey nonequilibrium quantum dynamics. The task of the quantum information theory investigates properties of information transmission by means of quantum channels. The von Neumann mutual information and Holevo capacity are useful entropic quantities for this purpose. This short note discusses the relation between the von Neumann mutual information and the Holevo function and also provides a useful formula for calculating these quantities. The result is applied to the quantum dense coding under the influence of an external environment, which is the generalization of the results of ref. 4 Suppose that Alice sends a signal state carrying classical information to Bob through a quantum channel described by an arbitrary trace-preserving completely positive map L. The signal states are represented by orthonormal quantum states j ki (k 1⁄4 1; 2; . . .) which are assumed to be realized with prior probabilities k (k 1⁄4 1; 2; . . .). Then the quantum state of Alice’s system, i.e., the input system of the quantum channel, is given by in 1⁄4 P k k ðkÞ in , where  ðkÞ in 1⁄4 j kih kj. In this case, we have the eigenvalue equation, inj ki 1⁄4 kj ki. After receiving the signal, Bob obtains the quantum state ðkÞ out 1⁄4 Lj kih kj with the probability k and thus the quantum state of the output system becomes out 1⁄4 Lin 1⁄4 P k k ðkÞ out. Then we consider the eigenvalue problems, outj ni 1⁄4 nj ni and ðkÞ outj ðkÞ n i 1⁄4 ðkÞ n j ðkÞ n i. To obtain the von Neumann mutual information between the input and output systems of the quantum channel L, it is necessary to introduce a composite state of the input and output systems, which is not uniquely determined. Since we now consider the transmission of classical information, we use a separable state in{out as a composite state, 5,6) in{out 1⁄4 X
In the present paper we introduce and study Bosonic Gaussian classical-quantum (c-q) channels; the embedding of the classical input into quantum is always possible and therefore the classical entanglement-assisted capacity Ceaunder appropriate input constraint is well defined. We prove a general property of entropy increase for weak complementary channel, that implies the equality C = Cea(where C is the unassisted capacity) for certain class of c-q Gaussian channels under appropriate energy-type constraint. On the other hand, we show by explicit example that the inequality Ceais not unusual for constrained c-q Gaussian channel.