Changepoint Problem in Quantumn Setting

TL;DR

This paper extends the classical changepoint detection problem to the quantum domain, analyzing the optimal measurement strategy for identifying a change in pure quantum states with minimal error.

Contribution

It introduces a quantum version of the changepoint problem, deriving the optimal measurement and error probabilities for distinguishing between two unknown pure states.

Findings

Derived the minimum average error probability for the quantum changepoint problem

Defined the optimal POVM for state discrimination in this context

Provided analytical expressions for error probabilities depending on state inner products

Abstract

In the changepoint problem, we determine when the distribution observed has changed to another one. We expand this problem to the quantum case where copies of an unknown pure state are being distributed. We study the fundamental case, which has only two candidates to choose. This problem is equal to identifying a given state with one of the two unknown states when multiple copies of the states are provided. In this paper, we assume that two candidate states are distributed independently and uniformly in the space of the whole pure states. The minimum of the averaged error probability is given and the optimal POVM is defined as to obtain it. Using this POVM, we also compute the error probability which depends on the inner product. These analytical results allow us to calculate the value in the asymptotic case, where this problem approaches to the usual discrimination problem.