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A Framework for Spatial Quantum Sensing

Luís Bugalho, Yasser Omar, Damian Markham

TL;DR

The paper presents a unified framework for spatial quantum sensing that treats distributed field estimation as linear inference problems over polynomial and analytic function models. By embedding interpolation, signal isolation, and least-squares within an algebraic-geometry toolkit, it derives exact and sufficient conditions for constructing linear estimators using generalized Vandermonde/alternant matrices and analyzes how sensor placement governs estimator accuracy. A central insight is that entangled, non-local sensor strategies can achieve superior precision for global linear functionals of the field, while the geometry of sensor placements yields error-free subspaces that are robust to model ambiguities. The framework is demonstrated through concrete examples (e.g., 2D grid interpolation and magnetic-field source finding) and provides a path toward scalable quantum sensing networks with practical implications from geophysics to biology.

Abstract

Analytical and algebraic geometry are valuable tools for dealing with problems involving analytical functions and polynomials. In what we connote as spatial quantum sensing the goal is, given an underlying field and a set of quantum sensors interrogating the field in a set of positions, to find an estimator for some property the field. This property can have multiple forms, be it distinguishing the source of a target signal, or evaluating the field (or a derivative thereof) in an arbitrary position. In this work we also link this problem to networks of quantum sensors, and the role and usefulness of entangling these sensors. We find that the estimators that come out as a solution to the problem are such that a non-local entangled strategy provides maximum precision. We start by working under the assumption of polynomial fields, which relates to the interpolation problem, and then generalize for any signal that is modeled via analytical functions, giving rise to any general least-squares estimator. We discuss the effects of the placement of the sensors in the estimation, namely, how to find well defined, construction error-free placements for the sensors. In the case of interpolation we provide concrete examples and proofs in a $m$-dimensional array of sensors, and discuss necessary and sufficient conditions for the more general cases. We provide clear examples of the possible use-cases and statements, and compare a non-local entangled strategy with the best local strategy for an interpolation problem, showing the benefit in terms of precision in a distributed sensing scenario. This is a key tool for a wide-range of problem in sensing problems, ranging from large-scale such as earth-sized experiments, to local-scale, such has biological experiments.

A Framework for Spatial Quantum Sensing

TL;DR

The paper presents a unified framework for spatial quantum sensing that treats distributed field estimation as linear inference problems over polynomial and analytic function models. By embedding interpolation, signal isolation, and least-squares within an algebraic-geometry toolkit, it derives exact and sufficient conditions for constructing linear estimators using generalized Vandermonde/alternant matrices and analyzes how sensor placement governs estimator accuracy. A central insight is that entangled, non-local sensor strategies can achieve superior precision for global linear functionals of the field, while the geometry of sensor placements yields error-free subspaces that are robust to model ambiguities. The framework is demonstrated through concrete examples (e.g., 2D grid interpolation and magnetic-field source finding) and provides a path toward scalable quantum sensing networks with practical implications from geophysics to biology.

Abstract

Analytical and algebraic geometry are valuable tools for dealing with problems involving analytical functions and polynomials. In what we connote as spatial quantum sensing the goal is, given an underlying field and a set of quantum sensors interrogating the field in a set of positions, to find an estimator for some property the field. This property can have multiple forms, be it distinguishing the source of a target signal, or evaluating the field (or a derivative thereof) in an arbitrary position. In this work we also link this problem to networks of quantum sensors, and the role and usefulness of entangling these sensors. We find that the estimators that come out as a solution to the problem are such that a non-local entangled strategy provides maximum precision. We start by working under the assumption of polynomial fields, which relates to the interpolation problem, and then generalize for any signal that is modeled via analytical functions, giving rise to any general least-squares estimator. We discuss the effects of the placement of the sensors in the estimation, namely, how to find well defined, construction error-free placements for the sensors. In the case of interpolation we provide concrete examples and proofs in a -dimensional array of sensors, and discuss necessary and sufficient conditions for the more general cases. We provide clear examples of the possible use-cases and statements, and compare a non-local entangled strategy with the best local strategy for an interpolation problem, showing the benefit in terms of precision in a distributed sensing scenario. This is a key tool for a wide-range of problem in sensing problems, ranging from large-scale such as earth-sized experiments, to local-scale, such has biological experiments.
Paper Structure (23 sections, 8 theorems, 54 equations, 3 figures)

This paper contains 23 sections, 8 theorems, 54 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Field Sensing
  3. Interpolation
  4. Taylor Expansion
  5. Expanding to the Derivatives of a Function
  6. Transforming the Expansion in a Linear Problem
  7. Interpolation Problem and Linear Combinations
  8. Error of Derivatives/Interpolation
  9. Example: Non-Local vs. Local Strategy on a 2D Grid Interpolation
  10. Analytical Functions
  11. Signal-Isolation Problem
  12. Example: Magnetic Field Source Finding
  13. Generalized Least-Squares
  14. Least-Squares Problem
  15. Invertibility, Pseudo-Invertibility and its consequences
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let $V(X,L)$ be a generalized Vandermonde matrix with $X$ a set of points in a $m$-dimensional space, and $L$ a set of different monomials in $m$ variables. $V(X,L)$ is invertible if and only if no polynomial spanned by $L$ vanishes for all $x\in X$.

Figures (3)

  • Figure 1: Schematic representation of the different parts of the problem setup. a) Mathematical model defined by a set of functions $\{f_1(x), f_2(x), \dots, f_k(x)\}$ over a domain $D$. The estimator for the field, or underlying function $\tilde{F}$ belongs to the linear span of the model functions $\mathcal{L}(\text{Model})$. b) All parties have access to the underlying field $F(x)$ at their own points in the domain $X = \{ x_1, x_2, \dots, x_j, \dots, x_p \} \in D$. In particular, they have access to some channel that encodes the outputs $\{ \Lambda_{F(x_1)},\Lambda_{F(x_2)}, \dots, \Lambda_{F(x_j)}, \dots, \Lambda_{F(x_p)} \}$.
  • Figure 2: Comparison between the best non-local and local strategies for the solution of a interpolation problem with $k=3$, using a equally spaced grid of sensors in two dimensions made of $3\times 3$ sensors (left) and $5\times 5$ sensors (right). To each point $\boldsymbol{x}_t = (x,y)$ we calculate and plot the gain on the precision (measured by the variance of the non-local divided by the variance of the local) of employing a non-local strategy to estimate $\hat{F}(\boldsymbol{x}_t)$.
  • Figure 3: Top Row: Comparison of the error of the interpolation of the polynomial $F(x) = (x-1)^3 + (y-1)^3$, considering the two possible equally spaced grids containing $3\times 3$ sensors (left) and $5\times 5$ sensors (right). Bottom Row: Comparison of the error of the interpolation of the polynomial $F(x) = (x-1)^5 + (y-1)^5$, considering the two possible equally spaced grids containing $3\times 3$ sensors (left) and $5\times 5$ sensors (right).

Theorems & Definitions (28)

  • Definition 3.1: Lower Set
  • Definition 3.2: Border and Cover
  • Example 3.1
  • Example 3.2
  • Theorem 3.1: Informal
  • Proposition 3.1
  • proof
  • Example 3.3: Transforming a set $X$ into a lower set $L$ in two dimensions
  • Definition 3.3: Labeling Function
  • Theorem 3.2
  • ...and 18 more