$d+1$ Measurement Bases are Sufficient for Determining $d$-Dimensional Quantum States: Theory and Experiment

TL;DR

This paper introduces a minimal measurement scheme requiring only d+1 bases for complete quantum state tomography in d-dimensional systems, validated experimentally on a silicon photonic chip for d=6.

Contribution

It presents a new theoretical method for quantum state reconstruction using fewer measurement bases, applicable even when mutually unbiased bases are absent.

Findings

Successfully reconstructed 6-dimensional quantum states experimentally.

Demonstrated that d+1 bases suffice for complete state characterization.

Provides a practical approach for high-dimensional quantum information tasks.

Abstract

A long-standing problem in quantum physics is to determine the minimal number of measurement bases required for the complete characterization of unknown quantum states, a question of particular relevance to high-dimensional quantum information processing. Here, we propose a quantum state tomography scheme that requires only $d+1$ projective measurement bases to fully reconstruct an arbitrary $d$-dimensional quantum state. As a proof-of-principle, we experimentally verified this scheme on a silicon photonic chip by reconstructing quantum states for $d=6$, in which a complete set of mutually unbiased bases does not exist. This approach offers new perspectives for quantum state characterization and measurement design, and holds promise for future applications in quantum information processing.