# 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.