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arXiv:quant-ph/9705054

On Quantum Communication Channels with Constrained Inputs

Alexander Semenovich Holevo

Published May 30, 1997• 16 citations
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Abstract

The purpose of this work is to extend the result of previous papers quant-ph/9611023, quant-ph/9703013 to quantum channels with additive constraints onto the input signal, by showing that the capacity of such channel is equal to the supremum of the entropy bound with respect to all apriori distributions satisfying the constraint. We also make an extension to channels with continuous alphabet. As an application we prove the formula for the capacity of the quantum Gaussian channel with constrained energy of the signal, establishing the asymptotic equivalence of this channel to the semiclassical photon channel. We also study the lower bounds for the reliability function of the pure-state Gaussian channel.

DOI
10.48550/arxiv.quant-ph/9705054

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The purpose of this work is to extend the result of previous papers quant-ph/9611023, quant-ph/9703013 to quantum channels with additive constraints onto the input signal, by showing that the capacity of such channel is equal to the supremum of the entropy bound with respect to all apriori distributions satisfying the constraint. We also make an extension to channels with continuous alphabet. As an application we prove the formula for the capacity of the quantum Gaussian channel with constrained energy of the signal, establishing the asymptotic equivalence of this channel to the semiclassical photon channel. We also study the lower bounds for the reliability function of the pure-state Gaussian channel.
The classical capacity of quantum channels is the tight upper bound for the transmission rate of classical information. This is a quantum counterpart of the foundational notion of the channel capacity introduced by Shannon. Bosonic Gaussian quantum channels provide a good model for optical communications. In order to properly define the classical capacity for these quantum systems, an energy constraint at the channel input is necessary, as in the classical case. A further restriction to Gaussian input ensembles defines the Gaussian (classical) capacity, which can be studied analytically. It also provides a lower bound on the classical capacity and moreover, it is conjectured to coincide with the classical capacity. Therefore, the Gaussian capacity is a useful and important notion in quantum information theory. Recently, we have shown that the study of both the classical and Gaussian capacity of an arbitrary single mode Gaussian quantum channel can be reduced to the study of a particular fiducial channel. In this work we consider the Gaussian capacity of the fiducial channel, discuss its additivity and analyze its dependence on the channel parameters. In addition, we extend previously obtained results on the optimal channel environment to the single mode fiducial channel. In particular, we show that the optimal channel environment for the lossy, amplification, and phase conjugating channels is given by a pure quantum state if its energy is constrained.
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.
This paper establishes a general theory of energy-constrained quantum and private capacities of quantum channels. We begin by defining various energy-constrained communication tasks, including quantum communication with a uniform energy constraint, entanglement transmission with an average energy constraint, private communication with a uniform energy constraint, and secret key transmission with an average energy constraint. We develop several code conversions, which allow us to conclude non-trivial relations between the capacities corresponding to the above tasks. We then show how the regularized, energy-constrained coherent information is equal to the capacity for the first two tasks and is an achievable rate for the latter two tasks, whenever the energy observable satisfies the Gibbs condition of having a well-defined thermal state for all temperatures and the channel satisfies a finite output-entropy condition. For degradable channels satisfying these conditions, we find that the single-letter energy-constrained coherent information is equal to all of the capacities. We finally apply our results to degradable quantum Gaussian channels and recover several results already established in the literature (in some cases, we prove new results in this domain). Contrary to what may appear from some statements made in the literature recently, proofs of these results do not require the solution of any kind of minimum output entropy conjecture or entropy photon-number inequality.
In the present paper we introduce and study Bosonic Gaussian classical-quantum (c-q) channels; embedding of the classical input in quantum input is always possible, and therefore the classical entanglement-assisted capacity C ea under an appropriate input constraint is well defined. We prove the general property of entropy increase for a weak complementary channel, which implies the equality C = C ea (where C is the unassisted capacity) for a certain class of c-q Gaussian channels under an appropriate energy-type constraint. On the other hand, we show by an explicit example that the inequality C < C ea is not unusual for constrained c-q Gaussian channel.