# Naimark extension for the single-photon canonical phase measurement

## TL;DR

This paper presents a method to implement the optimal measurement for distinguishing single-photon polarization states using Naimark extension and an optical network, enabling practical realization of quantum state discrimination.

## Contribution

It provides an analytical Naimark extension and a detection scheme for optimal single-photon polarization state discrimination with 2^N outcomes, advancing quantum measurement techniques.

## Key findings

- Derived an explicit Naimark extension for the POVM
- Proposed an optical implementation scheme for the measurement
- Achieved a scalable approach for multiple outcomes

## Abstract

We address the implementation of the positive operator-valued measure (POVM) describing the optimal M-outcomes discrimination of the polarization state of a single photon. Initially, the POVM elements are extended to projective operators by Naimark theorem, then the resulting projective measure is implemented by a Knill-Laflamme-Milburn scheme involving an optical network and photon counters. We find the analytical expression of the Naimark extension and the detection scheme that realise it for an arbitrary number of outcomes M = 2^N.