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IEEE Sarnoff Symposium

Novel geometrical solution to additivity problem of classical quantum channel capacity

László Gyöngyösi, Sándor Imre

Published Apr 1, 2010• 8 citations
Publisher

Abstract

Quantum channel additivity is currently an active area of research in Quantum Information Theory. The additivity conjectures have emerged from an attempt to find a closed form expression for the capacity of a noisy quantum channel. To this day, strict additivity for quantum channel capacity has been conjectured, but not proven. The open questions related to additivity of quantum channel capacities can be discussed by our novel approach. We show an efficient computational geometric method to analyze the additivity property, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as distance measure.

DOI
10.1109/sarnof.2010.5469715

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Quantum channel additivity is currently an active area of research in Quantum Information Theory. The additivity conjectures have emerged from an attempt to find a closed form expression for the capacity of a noisy quantum channel. To this day, strict additivity for quantum channel capacity has been conjectured, but not proven. The open questions related to additivity of quantum channel capacities can be discussed by our novel approach. We show an efficient computational geometric method to analyze the additivity property, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as distance measure.
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The amplitude damping quantum channel has great importance in practical quantum communications, since this channel model describes energy dissipation. The additivity of the amplitude damping quantum channel model is still an open question and an exciting subject of current research. We show an efficient information computational geometric method to analyze the additivity property of the amplitude damping quantum channel, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as a distance measure.
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:
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.
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 multiplicativity conjecture for quantum communication channels is formulated, validity of which for the values of parameter ppp close to 1 is related to the solution of the fundamental problem of additivity of the channel capacity in quantum information theory. The proof of the conjecture is given for the case of natural numbers ppp.