A Survey on Additivity Conjecture

Quantum material is one emerging branch of advanced materials. Quantum entanglement is one intrinsic property for quantum material, in particular, in quantum communication. Additivity conjecture is a long standing problem for quantum material to transmit information. This note surveys additivity conjecture in some kinds of forms, and introduces some known results including relations between them.

On Hastings' Counterexamples to the Minimum Output Entropy Additivity Conjecture

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.

ADDITIVITY CONJECTURE AND COVARIANT CHANNELS

In this survey paper we discuss the relation between the minimal output entropy and the χ-capacity for irreducibly covariant quantum channels implying equivalence of the additivity property for both quantities for such channels. The structure of the Weyl-covariant channels is described in detail.