How to combine three quantum states

TL;DR

This paper introduces a novel ternary operation for combining three quantum states, generalizing previous binary operations and providing a continuous interpolation between them using advanced group representation theory.

Contribution

It develops a new ternary operation for quantum states based on a unitary embedding of permutation groups, extending binary operations to a continuous family.

Findings

The ternary operation interpolates between nested binary operations.

The construction relies on a unitary version of Cayley's theorem.

Representation theory characterizes when permutation combinations are unitary.

Abstract

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. 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.