Multiphoton Tomography with Linear Optics and Photon Counting

TL;DR

This paper demonstrates that full quantum state tomography for multi-photon states can be achieved using only linear optics and photon counting, identifying the minimal measurement settings needed and proposing practical protocols.

Contribution

It establishes the minimal number of measurement bases for multi-photon state tomography using linear optics and photon counting, and introduces protocols based on unitary 2N-designs.

Findings

Minimal measurement settings are sufficient for state reconstruction.

Random linear optics configurations saturate the lower bound.

Unitary 2N-designs enable analytical state reconstruction protocols.

Abstract

Determining an unknown quantum state from an ensemble of identical systems is a fundamental, yet experimentally demanding, task in quantum science. Here we study the number of measurement bases needed to fully characterize an arbitrary multi-mode state containing a definite number of photons, or an arbitrary mixture of such states. We show this task can be achieved using only linear optics and photon counting, which yield a practical though non-universal set of projective measurements. We derive the minimum number of measurement settings required and numerically show that this lower bound is saturated with random linear optics configurations, such as when the corresponding unitary transformation is Haar-random. Furthermore, we show that for N photons, any unitary 2N-design can be used to derive an analytical, though non-optimal, state reconstruction protocol.