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

TL;DR

This paper characterizes linear maps on matrix spaces whose multiplicity maps reach maximal trace-norm growth, showing the transpose map is essentially unique in this regard, and applies this to quantum channel discrimination games.

Contribution

It proves the transpose map is uniquely characterized by maximal norm growth among linear maps, and applies this to identify unique quantum games with maximal entanglement advantage.

Findings

Transpose map achieves maximal norm growth uniquely.

Characterization of certain quantum channel discrimination games.

Werner-Holevo channels exemplify the maximal gap in entanglement advantage.

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.