Abstract
Given a linear map Φ : M_n → M_m, its multiplicity maps are defined as the family of linear maps Φ ⊗ id_k : M_n ⊗ M_k → M_m ⊗ M_k, where id_k denotes the identity on M_k. Let ·₁ denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e. Φ₁ = max{Φ(X)₁ : X ∈ M_n, X₁ = 1}. A fact of fundamental importance in both operator algebras and quantum information is that Φ ⊗ id_k₁ can grow with k. In general, the rate of growth is bounded by Φ ⊗ id_k₁ ≤ k Φ₁, 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.