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International Journal of Quantum Information

ADDITIVITY CONJECTURE AND COVARIANT CHANNELS

Alexander Semenovich Holevo

Published Dec 16, 2004• 41 citations
Publisher

Abstract

In this survey paper we discuss the relation between the minimal output entropy and the χ-capacity for irreducibly covariant quantum channels implying equivalence of the additivity property for both quantities for such channels. The structure of the Weyl-covariant channels is described in detail.
DOI
10.1142/s0219749905000530

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We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. This class is the one for which the norm of the output is maximized for an output state being a normalized projection. For such channels - which we characterize in their general form - we prove the additivity of the minimal output Renyi entropies with entropic parameters contained in [0, 2], generalizing an argument by Alicki and Fannes, and present several examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated.
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:
Additivity of minimal entropy output is proven for the class of quantum channels Λt(A):=tAT+(1−t)τ(A)Λ_t (A):=t A^{T}+(1-t)τ(A)Λt​(A):=tAT+(1−t)τ(A) in the parameter range −2/(d2−2)≤t≤1/(d+1)-2/(d^2-2)\le t \le 1/(d+1)−2/(d2−2)≤t≤1/(d+1).
We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. For this class of channels, the norm of the output is maximized for the output being a normalized projection. We prove the additivity of the minimal output Renyi entropies with entropic parameters α ∊ [0, 2], generalizing an argument by Alicki and Fannes, and present a number of examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify various instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated.