The transpose is the canonical example of a linear map on $M_n$ with maximum gap between its induced (trace-)norm and its completely bounded (trace-)norm. We prove that, up to a natural equivalence, the transpose is in fact the unique map with this property. Specifically, for a linear map $\Phi : M_n \rightarrow M_m$ with $\|\Phi\|_1 =1$, we show that its completely bounded trace-norm is $n$ if and only if $\Phi = \Psi T$, for $T$ the transpose on $M_n$ and $\Psi : M_n \rightarrow M_m$ a complete trace-norm isometry. We also provide characterizations of such maps, as well as complete trace-norm isometries in terms of properties of their Choi matrices; e.g. completely trace-norm isometric maps $M_n \rightarrow M_m$ are precisely the maps whose Choi matrices (with a certain normalization) are maximally entangled. Finally, we will describe an application of the uniqueness result to single-shot quantum channel discrimination in quantum information. The Werner-Holevo channels are a family of pairs of channels that satisfy a norm relation that, operationally, says that they can be perfectly discriminated using entanglement, but are hard to discriminate without entanglement. We prove that all channel discrimination games that satisfy this norm relation are essentially equivalent to the Werner-Holevo channel discrimination game.

This work is supported by OGS and the QEII-GSST.