Process-theoretic characterisation of the Hermitian adjoint

TL;DR

This paper provides an operational, process-theoretic characterization of the Hermitian adjoint in quantum theory, linking it to a fundamental physical principle and extending the concept to general process theories.

Contribution

It introduces a physical principle that fully characterizes the Hermitian adjoint in pure quantum theory and general process theories, giving it operational significance.

Findings

The adjoint associates states with tests for those states.

The principle induces a reflection operation in general process theories.

Provides an operational interpretation of the Hermitian adjoint.

Abstract

We show that the physical principle "the adjoint associates to each state a `test' for that state" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Also, we show that for general process theories, which all admit a diagrammatic representation, this physical principle induces a reflection operation.