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

Deepesh Singh, Ryan J. Marshman, Luis Villegas-Aguilar, Jens Eisert, Nora Tischler

TL;DR

The paper addresses the problem of proving no-go results for heralded linear-optical state generation by reformulating state-preparation tasks as polynomial-equation feasibility problems in the unknown linear map $A$ across $N$ modes. It adopts the Nullstellensatz Linear Algebra (NulLA) approach to produce infeasibility certificates, enabling rigorous lower bounds on required resources (e.g., input photons, heralding photons) for target states such as Bell and NOON states and for heralded gates like CNOT. The authors demonstrate that many practical tasks have infeasibility certificates at surprisingly low degrees, while worst-case bounds can be astronomically large, highlighting a practical asymmetry between theory and computation. The framework extends to probabilistic photon sources and multi-pair gate scenarios, offering a robust tool for resource budgeting and design constraints in photonic quantum technologies, with open-access MATLAB code to enable further exploration.

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.

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

TL;DR

The paper addresses the problem of proving no-go results for heralded linear-optical state generation by reformulating state-preparation tasks as polynomial-equation feasibility problems in the unknown linear map across modes. It adopts the Nullstellensatz Linear Algebra (NulLA) approach to produce infeasibility certificates, enabling rigorous lower bounds on required resources (e.g., input photons, heralding photons) for target states such as Bell and NOON states and for heralded gates like CNOT. The authors demonstrate that many practical tasks have infeasibility certificates at surprisingly low degrees, while worst-case bounds can be astronomically large, highlighting a practical asymmetry between theory and computation. The framework extends to probabilistic photon sources and multi-pair gate scenarios, offering a robust tool for resource budgeting and design constraints in photonic quantum technologies, with open-access MATLAB code to enable further exploration.

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.
Paper Structure (17 sections, 3 theorems, 16 equations, 2 figures, 1 table)

This paper contains 17 sections, 3 theorems, 16 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Technique to prove infeasibility
  3. Formulation of a heralded photonic state generation task as a system of polynomial equations
  4. Computing infeasibility certificates for heralded photonic state generation
  5. Properties of the technique
  6. Problem simplification
  7. Scaling
  8. Applications
  9. Resource bounds for Bell state generation
  10. Infeasibility certificate degrees for wider families of two-photon target states
  11. Inputs for vacuum-heralded NOON state generation
  12. Transformations for sets of input--output state pairs
  13. Example: Minimal resource requirements for the heralded CNOT gate
  14. Extension to probabilistic photon sources
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathbb{K}$ be an algebraically closed field. Given $f_1, f_2,\dots, f_s \in \mathbb{K}[x_1,\dots, x_n]$, the system of polynomial equations $f_1(x) = 0$, …, $f_s(x) = 0$, has no solution in ${\mathbb{K}}^n$ if and only if there exist polynomials $\beta_1,\dots, \beta_s \in \mathbb{K}[x_1,\dot

Figures (2)

  • Figure 1: Example demonstrating the optimality of single-photon inputs as described in Lemma \ref{['lem:optimal inputs']}. The scenario depicted in Fig. \ref{['fig:inp_actual']} involves state generation tasks requiring $n-3$ single photons in the top $n-3$ modes of an interferometer $\tilde{A}$, with two photons and one photon in the remaining two modes, respectively. We demonstrate that this setup can be replaced by the configuration shown in Fig. \ref{['fig:inp_optimal']}, where a larger interferometer $A$ achieves the generation of the same state using only single-photon inputs and heralding on vacuum in the newly introduced ancilla mode. The green pattern represents the desired configuration, while the red patterns indicate failed configurations that are ruled out by heralding.
  • Figure 2: Example demonstrating the optimality of the single-photon heralding pattern as described in Lemma \ref{['lem:optimal inputs']}. The scenario shown in Fig. \ref{['fig:out_actual']} involves state generation tasks requiring an input of $n$ single photons in the top $n$ modes of an interferometer $\tilde{A}$, with vacuum in $N-1-n$ modes. The dashed bottom mode below $\tilde{A}$ is there for illustration purposes when comparing with Fig. \ref{['fig:out_optimal']} only; it is superfluous for the state generation and can be disregarded. The desired outcome occurs when three photons occupy the last two modes, following a specific heralding pattern where one photon is in the second-to-last mode and two photons are in the last mode. This setup can be replaced by the one in Fig. \ref{['fig:out_optimal']}, where a larger interferometer $A$ achieves the generation of the same state using a single-photon heralding pattern. The green patterns indicate the possible photon-number measurement outcomes at the detectors that are associated with the successful heralding pattern of Fig. \ref{['fig:out_actual']}, with the first one being the optimal single-photon heralding pattern. The red patterns indicate the possible photon-number measurement outcomes at the detectors that are associated with the wrong heralding pattern of Fig. \ref{['fig:out_actual']}; these are successfully rejected by the single-photon heralding pattern.

Theorems & Definitions (3)

  • Theorem 1: Variant of Hilbert's Nullstellensatz
  • Lemma 1: Optimal input states and heralding patterns
  • Theorem 2: Infeasibility theorem