# Characterization of linear maps on $M_n$ whose multiplicity maps have maximal norm, with an application in quantum information. (arXiv:1710.03281v2 [quant-ph] UPDATED)

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.