Quantum hypothesis testing and sufficient subalgebras

TL;DR

This paper introduces a quantum analogue of classical sufficiency called 2-sufficiency, characterizing when a subalgebra contains all optimal tests for distinguishing quantum states, and explores its properties and equivalences.

Contribution

It defines 2-sufficiency for quantum states, linking it to classical sufficiency, and establishes conditions under which sufficiency and 2-sufficiency are equivalent in quantum tensor powers.

Findings

2-sufficiency contains all Bayes optimal tests

Sufficiency is equivalent to 2-sufficiency for tensor powers

Connections established between quantum and classical sufficiency

Abstract

We introduce a new notion of a sufficient subalgebra for quantum states: a subalgebra is 2- sufficient for a pair of states $\{\rho_0,\rho_1\}$ if it contains all Bayes optimal tests of $\rho_0$ against $\rho_1$. In classical statistics, this corresponds to the usual definition of sufficiency. We show this correspondence in the quantum setting for some special cases. Furthermore, we show that sufficiency is equivalent to 2 - sufficiency, if the latter is required for $\{\rho_0^{\otimes n},\rho_1^{\otimes}\}$, for all $n$.