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Entanglement in Markovian hybrid classical-quantum theories of gravity

Oliviero Angeli, Matteo Carlesso

TL;DR

This work analyzes entanglement generation in Gaussian Markovian master equations describing hybrid classical-quantum gravity. It proves that, for a broad class of Kossakowski matrices κ = κ2 ⊗ τ, positivity implies complete positivity in the partially transposed dynamics, enabling a PPT-based entanglement test with the criterion $\det \Re \kappa_2 < c^2$ when the entangling coupling is $h = c[1000]$, and extends this result to general $N$-mode partitions. The framework is then applied to two gravity models: a naïve classical-gravity toy model and the Tilloy-Diósi (TD) model, demonstrating that both can generate entanglement, with the root cause traced to nonlocality in the interaction. For the TD model, the entanglement condition yields a bound $d \lesssim 0.85 R_0$, implying that gravitationally induced entanglement experiments can falsify or strongly constrain $R_0$ and thus the model (e.g., $R_0$ bounds can improve by up to six orders of magnitude over current limits). Overall, the results provide a practical entanglement criterion for Gaussian hybrids and offer concrete experimental implications for tests of gravitational quantumness.

Abstract

Markovian master equations underlie many areas of modern physics and, despite their apparent simplicity, they encode a rich and complex dynamics which is still under active research. We identify a class of continuous variable Markovian master equations for which positivity and complete positivity become equivalent. We apply this result to characterize the positivity of the partially transposed evolution of bipartite Gaussian systems, which encodes the dynamics of entanglement. Finally, the entangling properties of models of classical gravity interacting with quantum matter are investigated in the context of the experimental proposals to detect gravitationally induced entanglement. We prove that entanglement generation can indeed take place within these models. We prove that entanglement generation can indeed take place within these models. In particular, by focusing on the Diósi-Penrose model for two gravitationally interacting masses, we show that entanglement-based experiments have the potential to either falsify the model entirely or constrain the free parameter of the model $R_0$ up to values six orders of magnitude above the current state of the art.

Entanglement in Markovian hybrid classical-quantum theories of gravity

TL;DR

This work analyzes entanglement generation in Gaussian Markovian master equations describing hybrid classical-quantum gravity. It proves that, for a broad class of Kossakowski matrices κ = κ2 ⊗ τ, positivity implies complete positivity in the partially transposed dynamics, enabling a PPT-based entanglement test with the criterion when the entangling coupling is , and extends this result to general -mode partitions. The framework is then applied to two gravity models: a naïve classical-gravity toy model and the Tilloy-Diósi (TD) model, demonstrating that both can generate entanglement, with the root cause traced to nonlocality in the interaction. For the TD model, the entanglement condition yields a bound , implying that gravitationally induced entanglement experiments can falsify or strongly constrain and thus the model (e.g., bounds can improve by up to six orders of magnitude over current limits). Overall, the results provide a practical entanglement criterion for Gaussian hybrids and offer concrete experimental implications for tests of gravitational quantumness.

Abstract

Markovian master equations underlie many areas of modern physics and, despite their apparent simplicity, they encode a rich and complex dynamics which is still under active research. We identify a class of continuous variable Markovian master equations for which positivity and complete positivity become equivalent. We apply this result to characterize the positivity of the partially transposed evolution of bipartite Gaussian systems, which encodes the dynamics of entanglement. Finally, the entangling properties of models of classical gravity interacting with quantum matter are investigated in the context of the experimental proposals to detect gravitationally induced entanglement. We prove that entanglement generation can indeed take place within these models. We prove that entanglement generation can indeed take place within these models. In particular, by focusing on the Diósi-Penrose model for two gravitationally interacting masses, we show that entanglement-based experiments have the potential to either falsify the model entirely or constrain the free parameter of the model up to values six orders of magnitude above the current state of the art.

Paper Structure

This paper contains 10 sections, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement in Markovian Master Equations
  3. Entanglement in models of classical gravity
  4. A naïve model for classical gravity
  5. Tilloy-Diósi model
  6. Conclusion
  7. Tools for continuous variables
  8. Partially transposed master equation
  9. Positivity and Complete Positivity
  10. Proof

Figures (3)

  • Figure 1: Gravitationally induced entanglement quantified by the logarithmic negativity ($E_N$). Blue line: unitary dynamics with the Newtonian potential alone. Orange line: naïve model \ref{['MAIN_EQ:NaiveME']} with $f = \sqrt{-V_\text{Newton}}$. Inset: relative difference of logarithmic negativity $\Delta E_N /E_N^\text{Newton} = (E_N^\text{Newton} - E_{N}^{\text{na\"ive}})/E_N^\text{Newton}$. The parameters used are those of Ref. krisnanda2020observable: $\omega=10^{5}\,$Hz, $m = 100\,\mu$g, $d = 0.3\,$mm.
  • Figure 2: Gravitationally induced entanglement quantified by the logarithmic negativity $E_N$ compared to the minimal value of $E_N$ that can be experimentally measured palomaki2013entangling (dashed orange line). Top panel:$E_N$ vs time. Blu line - Newtonian potential alone with the corresponding compatible values within a sensitivity of $10^{-2}$ (shaded area). Orange, green, and red lines - TD model with $R_0 = 3\,$mm ($d/R_0 = 0.1$), $R_0 = 0.5\,$mm ($d/R_0 = 0.6$) and $R_0 = 0.37\,$mm ($d/R_0 = 0.8$). Bottom panel:$E_N$ vs $R_0$. Different experimental times are considered: $t = 0.8$ s (blue line), $t = 4.7$ s (orange line) and $t = 10$ s (green line), $t = 25$ s (red line), $t = 50$ s (purple line). The parameters used are those of Ref. krisnanda2020observable: $\omega=10^{5}\,$Hz, $m = 100\,\mu$g, $d = 0.3\,$mm.
  • Figure 3: Logarithmic negativity ($E_N$) for the TD model for experimental times $t = 13$ s (blue line) and $t = 15$ s (orange line) compared against a speculative measured value of $E_N = 0.10\pm 0.01$ (gray shaded region). Only the values of $R_0$ for which the curves intersect the gray band would be compatible with the results. The parameters used are those of Ref. krisnanda2020observable: $\omega=10^{5}\,$Hz, $m = 100\,\mu$g, $d = 0.3\,$mm.