Quantum walk as a generalized measuring device

TL;DR

This paper demonstrates that a one-dimensional discrete-time quantum walk can be engineered to implement any rank 1 or 2 POVM on a single qubit, enabling generalized measurements like unambiguous state discrimination and SIC-POVM.

Contribution

It introduces a method to realize arbitrary rank 1 and 2 POVMs on a qubit using a quantum walk, expanding measurement capabilities in quantum information processing.

Findings

Quantum walks can implement any rank 1 and 2 POVM on a qubit.

Explicit constructions for unambiguous state discrimination and SIC-POVM.

Measurement outcomes correspond to particle positions in the quantum walk.

Abstract

We show that a one-dimensional discrete time quantum walk can be used to implement a generalized measurement in terms of positive operator value measure (POVM) on a single qubit. More precisely, we show that for a single qubit any set of rank 1 and rank 2 POVM elements can be generated by a properly engineered quantum walk. In such a scenario the measurement of particle at position x=i corresponds to a measurement of a POVM element E_i on a qubit. We explicitly construct quantum walks implementing unambiguous state discrimination and SIC-POVM.