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Russian Mathematical Surveys

The additivity problem for constrained quantum channels

Александр Семенович Холево, Максим Евгеньевич Широков

Published Apr 30, 2004• 0 citations
Publisher

Abstract

One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:

DOI
10.1070/rm2004v059n02abeh000729

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One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:
One of the main recent achievements of quantum information theory [1] is the proof of the equivalence of several (sub-, super-) additivity conjectures for finite-dimensional quantum channels and systems [2] (see the survey in [3]). The main purpose of this paper is to give a generalization of this result to the infinite-dimensional case. Let H and H ′ be separable Hilbert spaces. A quantum channel is a linear completely positive trace-preserving map Φ: T(H ) 7→ T(H ′), where T(H ) is the ideal of all trace class operators on H . In particular, Φ generates an affine map of the convex set S(H ) of states (that is, density operators) on the space H to the set S(H ′) of states on the space H ′ [1]. Important characteristics of a quantum channel are the output entropy HΦ(ρ) = H(Φ(ρ)), a lower-semicontinuous concave function on the input state space with values in [0,+∞], and its convex closure (see [4]), denoted by b HΦ(ρ) and called the b H-function of the channel Φ. It was shown in [5] that the convex closure of the output entropy of an arbitrary quantum channel Φ is given by the expression b HΦ(ρ) = inf μ Z
Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the χχχ-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the χχχ-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the χχχ-function and strong superadditivity of the entanglement of formation.
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the “classical capacity”, and the “maximal output purity”. All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the "classical capacity", and the "maximal output purity". All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture.
We develop an approximation approach to infinite dimensional quantum channels based on detailed investigation of the continuity properties of entropic characteristics of quantum channels and operations (trace-nonincreasing completely positive maps) as functions of a pair ``channel, input state''. The obtained results are then applied to the following problems: continuity of the χχχ-capacity as function of a channel; strong additivity of the χχχ-capacity for infinite dimensional channels; the analytical expression for the convex closure of the output entropy of arbitrary quantum channel.