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2020

The semiring of dichotomies and asymptotic relative submajorization

Perry, Christopher, Vrana, Péter, Werner, Albert H.

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We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings.

We find that an asymptotic variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum.
These strong constraints allow us to classify and explicitly describe all such monotones, leading to a rate formula expressed as an optimization involving sandwiched R\'enyi divergences.
As an application we give a new derivation of the strong converse error exponent in quantum hypothesis testing.

The semiring of dichotomies and asymptotic relative submajorization · Around

Built on

Resource theory of asymmetric distinguishability
Original
Xin Wang and Mark M Wilde · 1905
Earlier work this paper cites.
The radon-nikodym theorem for finite rings of operators
HA Dye · 1952
Earlier work this paper cites.
Conditional expectation in an operator algebra, iv (entropy and information)
Hisaharu Umegaki · 1962
Earlier work this paper cites.
von Neumann algebras
Jacques Dixmier · 1981
Earlier work this paper cites.
The asymptotic spectrum of tensors
Volker Strassen · 1988
Earlier work this paper cites.
The proper formula for relative entropy and its asymptotics in quantum probability
Fumio Hiai and Dénes Petz · 1991
Earlier work this paper cites.
A generalization of Strassen’s spectral theorem
Original
Péter Vrana · 2003
Earlier work this paper cites.
Strong converse and Stein’s lemma in quantum hypothesis testing
Tomohiro Ogawa and Hiroshi Nagaoka · 2005
Earlier work this paper cites.
Security of quantum key distribution
Renato Renner · 2008
Earlier work this paper cites.

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