How to combine three quantum states

We devise a ternary operation for combining three quantum states: it consists of permuting the input systems in a continuous fashion and then discarding all but one of them. This generalizes a binary operation recently studied by Audenaert et al. [arXiv:1503.04213] in the context of entropy power inequalities. In fact, our ternary operation continuously interpolates between all such nested binary operations. Our construction is based on a unitary version of Cayley's theorem: we use representation theory to show that any finite group can be naturally embedded in a continuous subgroup of the unitary group. Formally, this amounts to characterizing when a linear combination of certain permutations is unitary.