On the additivity hypothesis in quantum information theory

Grigori Gennadievich Amosov, Alexander Semenovich Holevo

A formulation of the multiplicativity hypothesis for quantum communication channels is given, the validity of which for parameter values~p, arbitrarily close to~1, implies a positive solution of the fundamental additivity problem for channel capacity in quantum information theory. A proof of this hypothesis is given for depolarizing channels in the case of integer~p.

Motohisa Fukuda

Janis Nötzel

We report a proof of the quantum Sanov theorem by the elementary application of basic facts about representations of the symmetric group, together with a complete characterization of the optimal error exponent in a situation where the null hypothesis is given by an arbitrarily varying quantum source instead. Our approach differs from previous ones in two ways. First, it supports a reasoning inspired by the 'method of types'. Second, the measurement scheme we propose to distinguish the two alternatives not only does that job asymptotically perfectly, but also yields additional information about the null hypothesis. An example of that is given. The measurement is composed of projections onto permutation-invariant subspaces, thus providing a direct link between one of the most basic tasks in quantum information on the one hand and fundamental objects in representation theory on the other. We additionally connect to the representation theory by proving a relation between Kostka numbers and quantum states, and to state estimation via a generalization of a well-known spectral estimation theorem to non-i.i.d. sequences.

Motohisa Fukuda

Nilanjana Datta, Alexander Semenovich Holevo, Yu. M. Suhov

Masahito Hayashi

Recently, it is well recognized that hypothesis testing has deep relations with other topics in quantum information theory as well as in classical information theory. These relations enable us to derive precise evaluation in the finite-length setting. However, such usefulness of hypothesis testing is not limited to information theoretical topics. For example, it can be used for verification of entangled state and quantum computer as well as guaranteeing the security of keys generated via quantum key distribution. In this talk, we overview these kinds of applications of hypothesis testing.