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Multiplicativity of \(p\)-norms of completely positive maps and the additivity problem in quantum information theory. (English. Russian original) Zbl 1325.81052

Russ. Math. Surv. 61, No. 2, 301-339 (2006); translation from Usp. Mat. Nauk 61, No. 2, 113-152 (2006).
Summary: The additivity problem is one of the most profound mathematical problems of quantum information theory. From an analytical point of view it is closely related to the multiplicative property, with respect to tensor products, of norms of maps on operator spaces equipped with the Schatten norms (non-commutative analogue of \(l_p\)-norms). In this paper we survey the current state of the problem.

MSC:

81P68 Quantum computation
94A15 Information theory (general)
81S25 Quantum stochastic calculus
46B28 Spaces of operators; tensor products; approximation properties
46L60 Applications of selfadjoint operator algebras to physics
46N50 Applications of functional analysis in quantum physics
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