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arXiv:quant-ph/0505022 · Journal of Physics A Mathematical and General

Extending additivity from symmetric to asymmetric channels

Motohisa Fukuda

Published Oct 26, 2005• 34 citations
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Abstract

We prove a lemma which allows one to extend results about the additivity of the minimal output entropy from highly symmetric channels to a much larger class. A similar result holds for the maximal output p -norm. Examples are given showing its use in a variety of situations. In particular, we prove the additivity and the multiplicativity for the shifted depolarizing channel.

DOI
10.1088/0305-4470/38/45/l02

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We prove a lemma which allows one to extend results about the additivity of the minimal output entropy from highly symmetric channels to a much larger class. A similar result holds for the maximal output p -norm. Examples are given showing its use in a variety of situations. In particular, we prove the additivity and the multiplicativity for the shifted depolarizing channel.
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In this paper we introduce a lemma. This lemma states that for a unitary covariant channel Psi\\PsiPsi and a channel MMM such that M(rho0)M(\\rho_0)M(rho0​) is of rank one for some state rho0\\rho_0rho0​ the additivity of the minimal output entropy of Psi\\PsiPsi implies that of PsicircM\\Psi \\circ MPsicircM, and the multiplicativity of the maximal output ppp-norm of Psi\\PsiPsi, that of PsicircM\\Psi \\circ MPsicircM. Applying the lemma to the depolarising channel we produce a new class of memoryless quantum channels for which the additivity of the minimal output entropy and the multiplicativity of the maximal output ppp-norm hold. This class includes the shifted depolarising channel.
We consider generalizations of depolarizing channels to maps in which the identity channel is replaced by a convex combinations of unitary conjugations. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this condition holds, but the channel has no evident covariance properties.
Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-norm. It then follows that the additivity and multiplicativity conjectures for these measures of optimal output purity hold for a product of channels if and only if they also hold for the product of their conjugates. This allows us to reduce these conjectures to the special case of maps taking M_d to M_d' with a minimal representation of dimension at most d. We find explicit expressions for the conjugates for a number of well-known examples, including entanglement-breaking channels, unital qubit channels, the depolarizing channel, and a subclass of random unitary channels. For the entanglement-breaking channels, channels this yields a new class of channels for which additivity and multiplicativity of optimal output purity can be established. For random unitary channels using the generalized Pauli matrices, we obtain a new formulation of the multiplicativity conjecture. The conjugate of the completely noisy channel plays a special role and suggests a mechanism for using noise to transmit information.
We consider generalizations of depolarizing channels to maps which are convex combinations of unitary conjugations (rather than the identity) and the completely noisy map. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this condition holds, but the channel has no evident covariance properties.