Manipulation of single-photon states encoded in transverse spatial modes: possible and impossible tasks

TL;DR

This paper investigates the limits of manipulating single-photon states in transverse spatial modes, proving Gaussian operations cannot arbitrarily modify higher-dimensional states but non-Gaussian operations can overcome this limit.

Contribution

It establishes the impossibility of arbitrary manipulation of qu$d$its with Gaussian operations and demonstrates how non-Gaussian operations can achieve full control, including realizing SU(3) algebra.

Findings

Gaussian operations cannot arbitrarily modify qu$d$its for d>2

Non-Gaussian operations enable overcoming the d=2 limit

Non-Gaussian operations realize the full SU(3) algebra

Abstract

Controlled generation and manipulation of photon states encoded in their spatial degrees of freedom is a crucial ingredient in many quantum information tasks exploiting higher-than-two dimensional encoding. Here, we prove the impossibility to arbitrarily modify $d$-level state superpositions (qu$d$its) for $d>2$, encoded in the transverse modes of light, with optical components associated to the group of symplectic transforms (Gaussian operations). Surprisingly, we also provide an explicit construction of how non-Gaussian operations acting on mode subspaces do enable to overcome the limit $d=2$. In addition, this set of operations realizes the full SU(3) algebra.