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arXiv:quant-ph/0410162 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Statistically constrained operator associated with additivity of communication channel

Toshio Fukumi

Published Jan 11, 2005• 0 citations
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Abstract

The power of quantum communication channel arises from additively of Holevo information, which certainly depends on the fact that the Hilbert space grows exponentially. The additively will be be shown to be achieved by stochastically maximizing a Hilbert space. In the conjecture we introduce operator statistics which is a natural extension of functions of operator, and employing it we discuss that the information is additive in probability.

DOI
10.1117/12.583213

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The power of quantum communication channel arises from additively of Holevo information, which certainly depends on the fact that the Hilbert space grows exponentially. The additively will be be shown to be achieved by stochastically maximizing a Hilbert space. In the conjecture we introduce operator statistics which is a natural extension of functions of operator, and employing it we discuss that the information is additive in probability.
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:
We study a natural generalization of the additivity problem in quantum information theory: given a pair of quantum channels, then what is the set of convex trace functions that attain their maximum on unentangled inputs if they are applied to the corresponding output state? We prove several results on the structure of the set of those convex functions that are ``additive'' in this more general sense. In particular, we show that all operator convex functions are additive for the Werner-Holevo channel in $3\ifmmode\times\else\texttimes\fi{}3$ dimensions, which contains the well-known additivity results for this channel as special cases.
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.