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

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.

💡 Research Summary

The paper addresses a fundamental problem in photonic quantum technologies: determining whether a desired discrete‑variable quantum state can be prepared using only linear‑optical networks, auxiliary photon detections (heralding), and a given set of input photons. Because linear optics lacks strong single‑photon nonlinearities, deterministic state preparation is impossible; instead one must rely on probabilistic, heralded schemes where a successful auxiliary measurement signals that the target state has been generated without destroying it. The central question—“does there exist a linear‑optical circuit that maps the chosen input to the target under the specified heralding pattern?”—can be translated into the feasibility of a system of polynomial equations.

The authors first recapitulate the algebraic formulation introduced in earlier work (Ref.