# An effective iterative method to build the Naimark extension of rank-n POVMs

## TL;DR

This paper introduces an improved iterative method for constructing the Naimark extension of rank-n POVMs, enabling efficient implementation of POVMs as projective measurements in larger Hilbert spaces.

## Contribution

The proposed method advances existing techniques by handling higher-rank POVMs and reducing computational complexity.

## Key findings

- Method successfully extends rank-n POVMs.
- Reduces computational steps compared to previous methods.
- Applicable to a broader class of POVMs.

## Abstract

We revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.