# Implementing positive-operator-valued-measurement elements in photonic circuits for performing minimum quantum state tomography of path qudits

## TL;DR

This paper proposes integrated photonic circuits for efficient minimum quantum state tomography of path qudits, utilizing symmetries and Naimark's theorem to optimize interferometer design for high-dimensional quantum states.

## Contribution

It introduces a novel method for designing compact photonic circuits for qudit tomography, generalizing to any dimension using equidistant states and symmetry principles.

## Key findings

- Smaller interferometers compared to existing methods
- Generalized scheme applicable to any qudit dimension
- Accurate determination of circuit parameters

## Abstract

Manipulation of qudits in optical tables is a difficult and nonscalable task. The use of integrated optical circuits opens new possibilities for the generation, manipulation, and characterization of high dimensional states besides the ease of transmission of these states through an optical fiber. In this work we propose photonic circuits to perform minimum quantum state tomography of path qudits and show how to determine all the constituents parameters of these circuits (beam splitters and phase shifters). Our strategies were based on the symmetries of the involved POVMs (positive operator-valued measures) suggested for minimum tomography and allowed us to obtain interferometers smaller than those obtained by other already known methods. The calculations of the transmittances and reflectivities of the beam splitters were made using the definition of probability operators in extended Hilbert spaces and the application of Naimark's theorem. The employment of equidistant states for the definition of the POVM elements allowed us to develop a recipe applicable to the tomography of qudits of any dimension, generalizing our scheme.