Five measurement bases determine pure quantum states on any dimension

TL;DR

This paper shows that any pure quantum state in any dimension can be uniquely reconstructed by measuring just five specific observables, simplifying high-dimensional quantum state tomography.

Contribution

It introduces a method requiring only five measurement bases for pure state reconstruction in any dimension, with linear scaling and robustness against errors.

Findings

Successfully reconstructed 8-dimensional quantum states experimentally

Demonstrated linear scaling of measurements with dimension

Method is robust and requires simple post-processing

Abstract

A long standing problem in quantum mechanics is the minimum number of observables required for the characterisation of unknown pure quantum states. The solution to this problem is specially important for the developing field of high-dimensional quantum information processing. In this work we demonstrate that any pure d-dimensional state is unambiguously reconstructed by measuring 5 observables, that is, via projective measurements onto the states of 5 orthonormal bases. Thus, in our method the total number of different measurement outcomes (5d) scales linearly with d. The state reconstruction is robust against experimental errors and requires simple post-processing, regardless of d. We experimentally demonstrate the feasibility of our scheme through the reconstruction of 8-dimensional quantum states, encoded in the momentum of single photons.