Characterization of linear maps onMnwhose multiplicity maps have maximal norm, with an application in quantum information

Abstract

Given a linear map Φ : M n → M m , its multiplicity maps are defined as the family of linear maps Φ ⊗ id k : M n ⊗ M k → M m ⊗ M k , where id k denotes the identity on M k . Let · 1 denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e. Φ 1 = max{ Φ(X) 1 : X ∈ M n , X 1 = 1}. A fact of fundamental importance in both operator algebras and quantum information is that Φ ⊗ id k 1 can grow with k. In general, the rate of growth is bounded by Φ ⊗ id k 1 ≤ k Φ 1 , and matrix transposition is the canonical example of a map achieving this bound. We prove that, up to an equivalence, the transpose is the unique map achieving this bound. The equivalence is given in terms of complete trace-norm isometries, and the proof relies on a particular characterization of complete trace-norm isometries regarding preservation of certain multiplication relations.We use this result to characterize the set of single-shot quantum channel discrimination games satisfying a norm relation that, operationally, implies that the game can be won with certainty using entanglement, but is hard to win without entanglement. Specifically, we show that the well-known example of such a game, involving the Werner-Holevo channels, is essentially the unique game satisfying this norm relation. This constitutes a step towards a characterization of single-shot quantum channel discrimination games with maximal gap between optimal performance of entangled and unentangled strategies.