Can Gravity Entangle Two Masses? First Decide Which Question You Mean

Can gravity entangle two masses?

I used to hear this as one question. It is at least three.

First, can a Newtonian quadratic interaction turn a separable Gaussian state into an entangled one? Second, if a calculation says yes, was gravity really the only non-local quantum resource in the model? Third, can an experiment prepare, preserve, and verify the required state before noise overwhelms it?

The answers need not agree. My current calculation finds “yes” to the first question, imposes a strict gravity-off test for the second, and finds an extraordinarily difficult set of requirements for the third. Mixing these answers together creates both false no-go theorems and false experimental optimism.

Status: this is an interactive account of ongoing undergraduate research supervised by Prof. Tomasz Paterek. The linked manuscript is a working draft, not a submitted or peer-reviewed paper. Exact statements below are scoped to the stated linearised two-mode Gaussian model.

Why Gaussian mechanics is the right first test

Consider two trapped masses, each moving along one coordinate. Put their positions and momenta into a phase-space vector

u=(x_A,p_A,x_B,p_B)^T.

A Gaussian state is fixed by its mean $\langle u\rangle$ and covariance matrix

\sigma_{ij}=\frac12\langle \Delta u_i\Delta u_j+\Delta u_j\Delta u_i \rangle.

That economy is the attraction. After expanding the interaction to second order, the Hamiltonian is quadratic and the covariance evolves by a matrix equation. There is no need to discretise a two-particle wavefunction over a large grid. Symplectic linear algebra carries the dynamics.

The economy also sets the boundary. Gaussian mechanics omits the cubic and higher terms that can create non-Gaussian states. It cannot answer every question about gravity and quantumness. It can answer a clean preliminary one: does the simplest controlled quantum model already permit entanglement?

For two modes, the entanglement test is especially sharp. Partial transpose amounts to reversing one momentum, $p_B\mapsto-p_B$ . Let $\tilde\nu_-$ be the smaller symplectic eigenvalue of the partially transposed covariance. Then

\tilde\nu_-<\frac{\hbar}{2}

certifies entanglement, and the base-two logarithmic negativity is

E_N=\max\!\left[0,-\log_2\!\left( \frac{2\tilde\nu_-}{\hbar}\right)\right].

A nonzero cross-covariance is not enough. Classical correlations also fill the off-diagonal blocks of $\sigma$ . The partial-transpose eigenvalue is what distinguishes correlation from entanglement in this setting, following the continuous-variable PPT criterion of Simon and the logarithmic negativity of Vidal and Werner.

Geometry decides whether gravity stiffens or softens

The same attractive force produces opposite quadratic signs in two common geometries. This surprised me at first.

Let the equilibrium centre separation be $L$ . If both masses move along the line joining them, their relative displacement changes the distance at first order. Expanding $-Gm^2/(L+r)$ gives a negative quadratic curvature: the relative mechanical mode softens. If the motion is transverse to the joining line, the distance is $\sqrt{L^2+r^2}$ and the quadratic curvature is positive: the mode stiffens.

The coordinate-independent check is the Hessian of a central potential. For $V=C/s^n$ ,

\partial_i\partial_j V =nCs^{-n-2}\big[(n+2)\hat s_i\hat s_j-\delta_{ij}\big].

For Newtonian gravity, $C=-Gm^2$ and $n=1$ . The Hessian has one longitudinal eigenvalue and two transverse eigenvalues with opposite sign and a factor of two between their magnitudes. “Gravity is attractive” does not tell us the sign of a fluctuation stiffness. The geometry does.

This check mattered when I reconstructed the fixed-coupling calculation in Poddubny et al.. I first reproduced the published dimensionless dynamics, then derived the microscopic coupling from the potential rather than trusting a sign convention. The distinction between reproducing a plotted model and validating its physical parameter map became a rule for the rest of the project.

A quench gives a constructive existence result

For equal masses and traps, centre-of-mass and relative coordinates diagonalise the quadratic Hamiltonian. In the collinear geometry the centre-of-mass mode keeps the bare trap frequency $\omega_m$ , while the relative frequency is

\Omega=\omega_m\sqrt{1-\kappa}, \qquad \kappa=\frac{4Gm}{L^3\omega_m^2}.

Stability requires $0<\kappa<1$ . Define the frequency mismatch

\gamma=\frac{\omega_m}{\Omega}=(1-\kappa)^{-1/2}.

Now begin with two independent local thermal states, each with mean occupation $\bar n$ , and switch on the gravitational curvature suddenly. The centre-of-mass covariance remains stationary. The relative covariance breathes because its initial width belongs to the old frequency. After one relative-mode quarter-period, the mismatch has rotated position squeezing into the correlations that partial transpose detects.

The peak logarithmic negativity is

E_N^{\mathrm{quench}} =\max\!\left[0,\log_2\!\left( \frac{\gamma}{2\bar n+1}\right)\right].

This is a constructive answer to the existence question. It specifies the initial covariance, the stable Hamiltonian, and the readout time. It is not the claim that an apparatus can execute the construction.

At $\kappa=0.5$ ,

E_N^{\mathrm{quench}} =\max\left[0,\frac12-\log_2(2\bar n+1)\right].

The state is entangled only if

\bar n<\frac{\sqrt2-1}{2}\simeq0.2071.

That number is easy to quote and easy to misunderstand. It is an occupation, not a laboratory temperature, and it belongs to this particular quench with $\kappa=0.5$ .

Gaussian quench calculator

How cold must the two local modes be?

0.362 bits

Hold the coupling fixed and raise the initial occupation. The quench does not fade gradually forever: it crosses a sharp Gaussian PPT threshold.

curvature fraction κ = 0.50weaksoft-mode wallinitial occupation n̄ = 0.05ground statethermal

coupled Gibbs0.112

sudden quench0.362

entanglement requires n̄ below0.207

readout time in trap periods0.354

frequency ratio γ = ωm/Ω1.414

current statePPT-entangled

Equal local thermal occupations and collinear quadratic gravity are assumed. κ must remain below one for stability. This calculator tests the analytic two-mode model; it does not supply a cooling, switching, or readout protocol.

The two bars compare different preparations. The coupled Gibbs state is the thermal equilibrium state of the already interacting Hamiltonian. The quench starts from a product thermal state and uses the subsequent breathing motion. At zero occupation the quench peak has twice the Gibbs logarithmic negativity. This is nonequilibrium gain, not free cooling.

Try holding $\kappa=0.5$ and moving $\bar n$ through $0.207$ . The displayed negativity reaches zero at the analytic PPT boundary. Then weaken $\kappa$ . The allowed occupation collapses, which is why a tiny gravitational curvature cannot be rescued by merely waiting longer.

The mass cancels where one might hope it helps

For two non-overlapping spheres of radius $a$ , density $\rho$ , and separation $L=2\lambda a$ , the largest available gravitational curvature has a simple form:

\omega_G^2=\frac{4Gm}{L^3} =\frac{2\pi G\rho}{3\lambda^3}.

The radius has disappeared. At fixed material density and fractional clearance, making both spheres larger increases the gravitational force and the inertia together. The characteristic curvature does not improve.

For fused silica near contact, the manuscript obtains $\omega_G/(2\pi)=88.26\,\mu\mathrm{Hz}$ . Reaching $\kappa=0.5$ therefore means lowering the final trap frequency to about $108.19\,\mu\mathrm{Hz}$ . The quarter-period readout then arrives after $3268$ seconds, or $54.5$ minutes. With $\bar n=0.05$ , the ideal model gives $E_N=0.3625$ bits, but the same occupation corresponds to an effective motional temperature of only $1.71\,\mathrm{fK}$ at that final frequency.

The femtokelvin number is not a refrigerator specification. A possible route would cool a much stiffer trap, decompress while preserving occupation, set the finite-clearance geometry, compensate the mean Newtonian force, and then quench. The current draft does not demonstrate a bounded, low-noise decompression schedule. That missing step separates the existence proof from an experimental protocol.

There is a second warning near $\kappa=1$ . The algebraic entanglement grows as the relative mode softens, but its spatial variance also grows. Eventually the wavepacket samples enough of the potential that the quadratic expansion fails. The stability wall is not a source of unlimited trustworthy entanglement.

Local feedback must pass a gravity-off test

Continuous measurement and feedback introduce another distinction: the state conditioned on a measurement record is not the unconditional state delivered when that record is ignored.

For a linear measurement, the conditional covariance obeys a Riccati equation. The Kalman filter estimates the conditional mean, and a linear-quadratic regulator chooses local forces. Across measurement records, however, those means fluctuate. The unconditional covariance has the form

\sigma^{u}=\sigma^{c}+\Xi, \qquad \Xi\ge0.

The additional positive covariance can only make the PPT test less favourable. Conditional entanglement is therefore an upper bound on the unconditional resource, not a finished experimental result.

This is also where a dangerous modelling shortcut appears. A dense feedback gain can look like a spring coupling mass $A$ to mass $B$ . If it is inserted directly into the operator drift without the measurement noise that carries the classical record, it becomes an unlabelled coherent non-local interaction. The simulation may then produce entanglement even with gravity removed.

That is a counterfeit success.

In a physically complete gravity-off calculation, local measurements, classical communication, and local conditional forces form an LOCC channel and cannot create entanglement. The numerical audit in the draft deliberately generates large classical cross-correlations under dense record feed-forward while retaining $E_N=0$ . Inserting the same gain as a noiseless Hamiltonian spring instead gives a false positive. This reproduces, in a concrete covariance model, the resource accounting emphasized by Kafri, Taylor and Milburn.

Periodic control revealed an error in my own reasoning

I also studied a periodically modulated separation. In Fourier language, a drive near the sum of two mode frequencies selects a two-mode-squeezing term, while a drive near their difference selects a passive beam-splitter term. The full covariance problem is periodic:

\dot\sigma=A(t)\sigma+\sigma A^T(t)+D.

Stability comes from the Floquet multipliers of the time-ordered monodromy. Once stable, the periodic covariance follows from a discrete Lyapunov equation. This gives an independent route alongside direct long-time integration.

An early version of my analysis suggested that the thermal tolerance diverged as the parametric drive approached its stability boundary. It was wrong. The formula I had used held the squeezed mode’s purity fixed while the open system continued to diffuse. In the corrected secular rotating-wave model, the threshold saturates at $\bar n=0.2071$ . A full calculation with a bare-local completely positive bath gives nearby but reservoir-dependent thresholds.

That correction changed how I use analytic reductions. Agreement with the closed-system limit is not enough. The reduced model must preserve the bath, noise, and purity assumptions of the full dynamics. The manuscript now checks Floquet exponents against direct covariance growth and checks periodic Lyapunov solutions against long-time integration.

The optimal-feedback work of Winkler et al. supplies the broader control setting. My contribution here is not a new general optimum. It is an audited comparison within declared control families, with gravity-off attribution and physicality tests applied before accepting a numerical point.

So, can gravity entangle the masses?

Inside the linearised Newtonian Gaussian model, yes. The stable finite-time quench is a direct construction, and its covariance passes the partial-transpose test. No measurement or non-local feedback is needed to create the ideal correlations.

Does the calculation hide another non-local resource? The gravity-off audit is designed to catch exactly that failure. Local record-based control stays separable; a noiseless cross-spring does not.

Can the construction be built now? The draft does not establish that. It asks for sub-phonon occupation after enormous decompression, tens of minutes of coherence at a microhertz trap frequency, high-efficiency measurement if feedback preparation is used, finite-range force compensation, and a force gradient budget below the Newtonian signal. Increasing the mass at fixed density does not evade the curvature ceiling.

The next useful calculation is consequently not another unconstrained search for a larger $E_N$ . It is a bounded decompression and readout protocol with explicit excursion, actuator, recoil, delay, and surface-force budgets. Only then do the three questions begin to share one answer.