Experimental Observation of Purity-Like Invariants of Multi-photon States in Linear Optics

TL;DR

This paper introduces a new Hermitian transfer matrix approach to identify and experimentally verify invariants of multi-photon states in linear optical networks, enhancing understanding of their evolution and constraints.

Contribution

It presents a novel method to decompose multi-photon state purity into invariants and experimentally demonstrates their conservation in linear optics.

Findings

Purity decomposes into three invariants from single-photon and multi-photon interference.

Experimental verification confirms invariants are conserved during LON evolution.

Provides insights into the intrinsic constraints of multi-photon state manipulation.

Abstract

Linear optical networks (LONs) with multi-photon inputs offer a powerful platform for advanced quantum technologies. However, the number of degrees of freedom of a LON is far fewer than the dimensionality of the multi-photon multi-mode Fock space, therefore it cannot implement arbitrary unitary evolutions on multi-photon states. Understanding these intrinsic constraints is essential for the preparation, manipulation, and measurement of multi-photon states with LONs. Although several properties of the multi-photon state have been shown to be invariant under LON unitary evolution, their physical interpretation remains elusive. Here, we introduce a Hermitian transfer matrix approach to explore the multi-photon evolution, revealing that the overall state purity decomposes into three distinct invariants -- each arising from either single-photon dynamics or the multi-photon interference. We experimentally observe these purity-like invariants by preparing distinct initial states, applying LON unitaries, and measuring the resulting invariants. Our results not only confirm their conservation but also provide valuable insights into multi-photon state evolution in linear optics.