Quantum hypothesis testing between qubit states with parity

TL;DR

This paper studies quantum hypothesis testing between qubit states under parity symmetry constraints, providing a complete characterization of error probabilities and the number of copies needed for perfect discrimination.

Contribution

It introduces a framework for asymmetric quantum hypothesis testing with parity-invariant measurements and characterizes the asymptotic error behavior for pure and mixed states.

Findings

Minimal error probability reduces to zero with finite copies if states differ after parity twirling.

Derived the critical number of copies needed for zero error probability.

Extended analysis to include maximally mixed states in the discrimination task.

Abstract

Quantum hypothesis testing (QHT) provides an effective method to discriminate between two quantum states using a two-outcome positive operator-valued measure (POVM). Two types of decision errors in a QHT can occur. In this paper we focus on the asymmetric setting of QHT, where the two types of decision errors are treated unequally, considering the operational limitations arising from the lack of a reference frame for chirality. This reference frame is associated with the group $\bbZ_2$ consisting of the identity transformation and the parity transformation. Thus, we have to discriminate between two qubit states by performing the $\bbZ_2$-invariant POVMs only. We start from the discrimination between two pure states. By solving the specific optimization problem we completely characterize the asymptotic behavior of the minimal probability of type-II error which occurs when the null hypothesis is accepted when it is false. Our results reveal that the minimal probability reduces to zero in a finite number of copies, if the $\bbZ_2$-twirlings of such two pure states are different. We further derive the critical number of copies such that the minimal probability reduces to zero. Finally, we replace one of the two pure states with a maximally mixed state, and similarly characterize the asymptotic behavior of the minimal probability of type-II error.