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Summary: arXiv:1003.4925v1[quant-ph]25Mar2010
HASTINGS' ADDITIVITY COUNTEREXAMPLE VIA DVORETZKY'S
THEOREM
GUILLAUME AUBRUN, STANISLAW SZAREK, AND ELISABETH WERNER
Abstract. The goal of this note is to show that Hastings' counterexample to the additivity
of minimal output von Neumann entropy can be readily deduced from a sharp version of
Dvoretzky's theorem.
Introduction
A fundamental problem in Quantum Information Theory is to determine the capacity
of a quantum channel to transmit classical information. The seminal HolevoSchumacher
Westmoreland theorem expresses this capacity as a regularization of the so-called Holevo
-quantity (which gives the one-shot capacity) over multiple uses of the channel; see, e.g.,
[15]. This extra step could have been skipped if the -quantity had been additive, i.e., if
(1) ( ) = () + ()
for every pair (, ) of quantum channels. It would have then followed that the -quantity and
the capacity coincide, yielding a single-letter formula for the latter. Determining the veracity
of (1) had been a major open problem for at least a decade (we refer, e.g., to the survey [11]).
A substantial progress was made by Shor [18] who showed that (1) was formally equivalent to
the additivity of the minimal output von Neumann entropy of quantum channels -- a much
more tractable quantity. Using this equivalence, the equality (1) was eventually shown to be
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