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

Abstract

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

Accepted by Quantum

References in corpus (3)

• All entangled states are useful for channel discrimination
• Multivariate Trace Inequalities
• Ancilla dimension in quantum channel discrimination

Cited by in corpus (1)

• Completely Bounded Norms of $k$-positive Maps