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Equivalence of additivity questions in quantum information theory. (English) Zbl 1070.81030

The study of quantum information theory has led to a number of seemingly related open questions that center around whether certain quantities are additive. In this paper it is shown that four of these questions are equivalent, namely
(1)
additivity of the minimum entropy output of a quantum channel,
(2)
additivity of the Holevo capacity of a quantum channel,
(3)
additivity of the entanglement of formation,
(4)
strong superadditivity of the entanglement of formation,
are either true or false for all quantum channels.

Some of the implications were known before. This paper both proves the missing (among them most difficult) implications and summarizes the previously known ones. The principal new tool in the proofs is an ingeniuos channel extension construction that allows to reduce the additivity of capacity for linearly constrained channels (and hence, of entanglement of formation) to the additivity of unconstrained extensions of the channels (Sec.4; also Sec. 7).
Two of the other basic ingredients are an observation of Matsumoto, Shimono and Winter (quant-ph/0206148) that the Stinespring dilation theorem relates a constrained version of the Holevo capacity formula to the entanglement of formation (Sec. 3). The second is the realization that the entanglement of formation is a linear programming problem, and so there is also a dual linear formulation, noted independently by the author and by Audenaert and Braunstein (quant-ph/0303045) (Sec. 5). In Sec. 6 it is shown that additivity of entanglement of formation implies strong superadditivity of entanglement of formation, an implication independently and somewhat later discovered by Pomeransky (quant-ph/0305056).

MSC:

81P15 Quantum measurement theory, state operations, state preparations
81S25 Quantum stochastic calculus
94A15 Information theory (general)
81P68 Quantum computation
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