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arXiv:quant-ph/0212057 · Quantum Information and Computation

Maximal p-norms of entanglement breaking channels

Christopher King

Published Mar 1, 2003• 68 citations
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Abstract

It shown that when one of the components of a product channel is entanglement breaking, the output state with maximal p-norm is always a product state. This result complements Shor's theorem that both minimal entropy and Holevo capacity are additive for entanglement breaking channels. It is also shown how Shor's results can be recovered from the p-norm results by considering their behavior for p close to one.

DOI
10.26421/qic3.2-9

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•Jan 1, 2002•157 citations
It shown that when one of the components of a product channel is entanglement breaking, the output state with maximal p-norm is always a product state. This result complements Shor's theorem that both minimal entropy and Holevo capacity are additive for entanglement breaking channels. It is also shown how Shor's results can be recovered from the p-norm results by considering their behavior for p close to one.
It is known that the minimal output entropy is additive for any product of entanglement breaking (EB) channels. The same is true for the Renyi entropy, where additivity is equivalent to multiplicativity of the \norm{1}{q} norm for all q≥1q \ge 1q≥1. In this paper we consider the related question of multiplicativity of the \norm{2}{q} norm for q>2q > 2q>2 for entanglement breaking channels. We prove that multiplicativity holds in this case for certain classes of EB channels, including both the CQ and QC channels.
We consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely, that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps on n-qubit states, including all those for which at least one map is unital. For the class of unital channels on C/sup 2/, we show that additivity of minimal entropy implies that the Holevo (see IEEE Trans. Inform. Theory, vol.44, p.269-73, 1998 and Russian Math. Surv., p.1295-1331, 1999) capacity of the channel is additive over two inputs, achievable with orthogonal states, and equal to the Shannon capacity. This implies that superadditivity of the capacity is possible only for nonunital channels.
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.
We show that for the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the Holevo-Schumacher-Westmoreland capacity are additive. In addition, for the tensor product of two arbitrary quantum channels, we give a bound involving entanglement of formation for the amount of subadditivity (for minimum entropy output) or superadditivity (for classical capacity) that can occur. One of the more important open questions of quantum information theory is the determination of the capacity of a quantum channel for carrying classical information. This question has been only partially resolved. If entanglement between multiple inputs to the channel is not allowed, a formula for the classical capacity of a quantum channel has indeed been discovered [1, 2]. This capacity formula for a quantum channel Ψ is χ ∗ (Ψ) = max p i,ρ i H Ψ(piρi) − ∑