Constructing a qubit POVM from quantum data

TL;DR

This paper explores methods for constructing a POVM to distinguish between two unknown pure qubit states within an ensemble, using quantum data and varying prior knowledge about the states' properties.

Contribution

It introduces approaches for learning a POVM from quantum data under different assumptions about the states' Bloch vectors and prior probabilities.

Findings

Effective POVMs can be constructed with partial state information.

Different prior probability scenarios affect the POVM design.

The methods enable state discrimination without complete state knowledge.

Abstract

Given an ensemble of qubits, which we are told consists of a mixture of two pure states, one with probability $\eta_{0}$ and one with probability $\eta_{1}$, we want to find a POVM that will discriminate between the two states by measuring the qubits. We do not know the states, and for any given qubit, we do not know which of the two states it is in. This can be viewed as learning a POVM from quantum data. Once found, the POVM can be used to separate the remaining qubits in the ensemble into two groups, corresponding to the two states present in the ensemble. In order to find the POVM, we need more information about the possible states. We examine several cases. First, we suppose that we know that the Bloch vectors of the states lie in the x-z plane and their \emph{a priori} probabilities are equal. We next keep the restriction to the x-z plane, but allow the \emph{a priori} probabilities to be different. Finally, we consider the case in which the Bloch vectors of the states have the same z component.