Entanglement without Quantum Mechanics: Operational Constraints on the Quantum Signature

TL;DR

Entanglement is not uniquely quantum; under restricted measurements, classical correlations can mimic entanglement when reformulated in Hilbert-space language. The authors formalize an operational hierarchy using the Wigner–Weyl mapping to separate representational artifacts, classically reproducible nonseparability, and genuine quantum entanglement, with positivity and Wigner negativity serving as key boundaries. They illustrate with Gaussian mixtures and beamsplitter transformations that standard covariance tests can misclassify, underscoring the need for full positivity checks and nonclassicality signatures to certify true entanglement. The work clarifies the classical-quantum boundary and informs interpretations of experiments on entanglement and gravity-related correlations.

Abstract

Entanglement is often regarded as an inherently quantum feature. We show that this does not have to be the case: under restricted operational access, classical correlations can appear nonseparable when expressed in the formalism of quantum mechanics. If an observer is limited to a constrained set of measurements and transformations, certain classical phase-space distributions can mimic entanglement-like behaviours. Imposing positivity of the associated Hilbert space operator as a physicality requirement removes some of these representational artifacts, revealing a regime in which nonseparability is genuine but still reproducible by classical models. Only when the operational restrictions on the observer are lifted further--allowing operational tests of measurement incompatibility or other nonclassical signatures--does one obtain entanglement that can no longer be captured by any classical description. This operational hierarchy distinguishes classical artifacts, classically reproducible nonseparability, and genuine entanglement.

Figures (4)

• Figure 1: Overlap of classical and quantum state spaces in the Wigner--Weyl representation. States in the intersection are operationally indistinguishable when access is restricted to phase-space (quadrature) statistics.
• Figure 2: Smallest symplectic eigenvalues of the covariance matrix $\Sigma(d)$ as a function of displacement $d$ (with $\hbar=1$). The RS bound (dashed line) certifies physicality, while violation of PPT (green line below $1/2$) would normally indicate entanglement. In the highlighted region, the covariance suggests a valid entangled state, but the underlying operator is non-positive, illustrating representational entanglement. Covariance-based analysis would misdiagnose entanglement, but the operator spectrum reveals non-positivity. Parameter values to generate these curves are given in Appendices \ref{['app:covariance']} and \ref{['app:negativity']}.
• Figure 3: Three regimes of nonseparability: (i) RE: representational entanglement (non-positive), (ii) HE: hybrid entanglement (classically reproducible), and (iii) GE: genuine entanglement (quantum-only).
• Figure 4: Smallest eigenvalue of the Weyl-transformed kernel $K(x,x')$ for the displaced Gaussian mixture $P(z)$ of Eq. \ref{['eq:Gaussian_mixture']}, evaluated on a discretized position grid. For all displacements $d$, the kernel exhibits negative eigenvalues, confirming non-positivity of the corresponding Hilbert-space operator and thus the non-physical nature of the apparent entanglement.