Characterization of linear maps on Mn whose multiplicity maps have maximal norm, with an application in quantum information
Daniel Puzzuoli
Given a linear map Φ:Mn→Mm , its multiplicity maps are defined as the family of linear maps Φ⊗idk:Mn⊗Mk→Mm⊗Mk , where idk denotes the identity on Mk . Let ‖⋅‖1 denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e. ‖Φ‖1=max{‖Φ(X)‖1:X∈Mn,‖X‖1=1} . A fact of fundamental importance in both operator algebras and quantum information is that ‖Φ⊗idk‖1 can grow with k . In general, the rate of growth is bounded by ‖Φ⊗idk‖1≤k‖Φ‖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.
arXiv:1710.03281•Quantum•Feb 5, 2018•1 citation