Characterization of linear maps on $M_n$ whose multiplicity maps have maximal norm, with an application in quantum information

Abstract: Given a linear map $\Phi : M_n \rightarrow M_m$, its multiplicity maps are defined as the family of linear maps $\Phi \otimes \text{id}_k : M_n \otimes M_k \rightarrow M_m \otimes M_k$, where $\text{id}_k$ denotes the identity on $M_k$. Let $|\cdot|_1$ denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e. $|\Phi|_1 = \max{|\Phi(X)|_1 : X \in M_n, |X|_1 = 1}$. A fact of fundamental importance in both operator algebras and quantum information is that $|\Phi \otimes \text{id}_k|_1$ can grow with $k$. In general, the rate of growth is bounded by $|\Phi \otimes \text{id}_k|_1 \leq k |\Phi|_1$, 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.