Quantum Advantage for Sensing Properties of Classical Fields

Abstract

Modern precision experiments often probe unknown classical fields with bosonic sensors in quantum-noise-limited regimes where vacuum fluctuations limit conventional readout. We introduce Quantum Signal Learning (QSL), a sensing framework that extends metrology to a broader property-learning setting, and propose a quantum-enhanced protocol that simultaneously estimates many properties of a classical signal with shot noise suppressed below the vacuum level. Our scheme requires only two-mode squeezing, passive optics, and static homodyne measurements, and enables post-hoc classical estimation of many properties from the same experimental dataset. We prove that our protocol enables a quantum speedup for common classical sensing tasks, including measuring electromagnetic correlations, real-time feedback control of interferometric cavities, and Fourier-domain matched filtering. To establish these separations, we introduce an optimal-transport conditioning method, and show both worst-case exponential separations from all entanglement-free strategies and practical speedups over homodyne and heterodyne baselines. We further show that when squeezing is treated as a resource, a protocol with squeezed light can sense a structured classical background exponentially faster than any coherent classical probe.

Figures (5)

• Figure 1: (a) Quantum-enhanced signal learning vs. conventional electromagnetic (EM) sensing. Bell QSL leverages probes entangled with quantum memory to collect a classical record that is used for quantum-limited, informationally-complete estimation of the EM field distribution. Heterodyne measurements suffer a vacuum noise penalty, blurring fine-grained features, while homodyne measurements only resolve marginals that may not contain higher-dimensional properties. (b) Quantum advantage for estimating correlations in an electromagnetic field. Minimum sample complexity required to estimate a Gaussian EM field covariance $c$ in the shot-noise-limited regime, as in \ref{['eq:Sigma_c']}. At $r=2$ and $|\sigma_x^2 - \sigma_p^2| \ll c$, in the presence of white noise with amplitude $0.2 c$, Bell QSL scales polynomially and orders of magnitude below the vacuum noise limit, achieving quantum advantage over both homodyne and heterodyne tomography.
• Figure 1: Measurement distributions corresponding to peaked displacement channels in \ref{['eq:prac_eps_discrete_hyps']}. $P_{\mathrm{diag}}$ is depicted in red and $Q_\varepsilon$ in green. The homodyne marginals upon measuring along axes $\{0, \pi/2\}$ are shown to the right. Their only distinguishable feature is a slight displacement in means along the $p$ axis.
• Figure 2: (a) Conventional bosonic sensing of classical fields. Homodyne measurements project high-dimensional field distributions onto chosen marginal axes, losing information, while heterodyne measurements blur sharp features with vacuum noise. (b) Bell QSL. Entangled two-mode squeezed vacuum probes are prepared, with half the modes acting as probes and the other half stored in quantum memory. Beamsplitters and homodyne measurements constitute CV Bell-basis measurement, enabling informationally complete, sharp property recovery.
• Figure 3: The problem wedge of quantum advantage, illustrated for the delta-mixture toy example. As the symmetry-breaking parameter $\varepsilon \rightarrow 0$, homodyne shots become non-identifiable and sample complexity diverges. In the red region, $\varepsilon$ is exponentially small relative to $b$, yielding exponential advantage, and in the orange region, it is polynomially small. The same scaling controls the advantage in Theorem \ref{['thm:EM_advantage_informal']}, with $\varepsilon\rightarrow\Delta$.
• Figure 4: Classical unsupervised learning employing Bell QSL data can learn signal properties beyond the scope of heterodyne and homodyne tomography. Each measurement method collects many draws from the underlying distributions $P, Q$ from \ref{['eq:deltas_eq_main_text']}, with squeezing $r=1.2$, $a=0.35$, $b = 0.06$, and $\varepsilon = 0$. Each dataset is fed into unsupervised ML (Gaussian kernel PCA) to learn whether each shot was drawn from $P$ (sign +1) or $Q$ (sign -1). Even at modest $b<1$ and squeezing, and with appreciably large $\epsilon$, the classical Bell dataset constitutes a far more feature-rich training dataset for ML algorithms than conventional strategies.

Theorems & Definitions (75)

• Definition 1: Quantum Signal Learning
• Theorem 1: Practical quantum advantage in learning an EM correlation, informal
• Theorem 2: Worst-case quantum advantage for matched filtering, informal
• Theorem 3: OT ambiguity implies minimax lower bounds, informal
• Theorem 4: Exponential quantum advantage with only squeezed probes
• proof
• proof
• proof
• Proposition 1: Entanglement-enabled Bell sampling as low-noise two-dimensional readout
• proof