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. [ADO16] in the context of entropy power inequalities. Our ternary operation continuously interpolates between all such nested binary operations. Our construction is based on a unitary version of Cayley’s theorem: using representation theory we show that any finite group can be naturally embedded into a continuous subgroup of the unitary group. Formally, this amounts to characterizing when a linear combination of certain permutations is unitary.