# Obtaining conclusive information from incomplete experimental quantum tomography

## TL;DR

This paper shows that incomplete quantum tomography can conclusively identify the subset of the state space a quantum system belongs to, using specific measurements and partitioning strategies, demonstrated experimentally on two-photon states.

## Contribution

It introduces a method to solve membership problems in quantum state space using incomplete measurements, with experimental validation on two-photon polarization states.

## Key findings

- Conclusive identification of state space subsets with incomplete measurements
- Development of measurement strategies for membership problems
- Experimental demonstration on two-photon polarization states

## Abstract

We demonstrate that incomplete quantum tomography can give conclusive information in experimental realizations. We divide the state space into a union of multiple disjoint subsets and determine conclusively which of the subsets a system, prepared in completely unknown state, belongs to. In other words we construct and solve membership problems. Our membership problems are partitions of the state space into a union of four disjoint sets formed by fixing two maximally entangled reference states and boundary values of fidelity function "radius" between the reference states and the unknown preparation. We study the necessary and sufficient conditions of the measurements which solve these membership problems conclusively. We construct and experimentally implement such informationally incomplete measurement on two-photon polarization states with combined one-qubit measurements, and solve the membership problem in example cases.