Rigorous no-go theorems for heralded linear-optical state generation tasks

TL;DR

This paper introduces a rigorous algebraic approach using Nullstellensatz Linear Algebra to determine the feasibility of photonic quantum state preparation tasks, providing definitive no-go results and resource bounds.

Contribution

It applies algebraic geometry techniques to quantum optics, enabling conclusive infeasibility proofs for linear-optical state generation tasks.

Findings

Validated and established lower bounds for optical state and gate realization.

Provided a systematic method to prove infeasibility of certain quantum state preparations.

Demonstrated the approach's effectiveness in complex photonic quantum systems.

Abstract

A major challenge in photonic quantum technologies is developing strategies to prepare suitable discrete-variable quantum states using simple input states, linear optics, and auxiliary photon measurements to identify successful outcomes. Fundamentally, this challenge arises from the lack of strong non-linearities on the single-photon level, meaning that photonic state preparation based on linear optics cannot benefit from the deterministic gate-based approach available to other physical platforms. Instead, the preparation of quantum states can be probabilistically implemented using single photons, linear-optical networks, and photon detection. However, determining whether an input state can be transformed into a target state using a specific measurement pattern - a problem that can be mapped to deciding the feasibility of a system of polynomial equations - is a complex problem in general. To solve it, we apply the Nullstellensatz Linear Algebra algorithm from algebraic geometry to quantum state generation; this can provide definitive no-go results by proving infeasibility when the state preparation task in question has no solution. We demonstrate this capability to validate and establish lower bounds on the physical resource requirements for the realization of several ubiquitous optical states and gates.