Pure states are optimal for total energy-constrained quantum channel discrimination

Abstract

We consider the discrimination of two quantum channels acting on bosonic modes, with a probe state of fixed mean photon number and no ancilla. We prove that pure probes are optimal: for every mixed probe there is a pure probe with exactly the same mean photon number that performs at least as well. The result holds for arbitrary channels and for every figure of merit built from the Chernoff quantities $\Tr[ρ_1^sρ_2^{1-s}]$, including the quantum Chernoff exponent, as well as for the single-shot error probability. The proof relies on a decomposition of any state of mean photon number $\nb$ into pure states of mean photon number exactly $\nb$, which extends a matrix rank-one decomposition of Huang and Zhang to density operators of infinite rank.