1 Introduction

Two masses, each in a spatial superposition, become entangled through their Newtonian interaction if that interaction is carried by a quantum mediator . A classical channel acting by local operations and classical communication cannot entangle them, so a positive entanglement witness bears on the quantum nature of gravity . The QGEM programme has turned the proposal into explicit designs: the original linear scheme , and screened and trapped variants that bring the masses closer behind a conducting plate .

The LOCC argument has two known qualifications. Classical gravity coupled to quantum matter can produce entanglement in some theories , and the Diósi–Penrose (DP) model with a classical, smeared Newtonian feedback entangles two bodies when the smearing length exceeds their separation . A forecast must therefore state which classical theory it tests, and at which smearing length.

This paper forecasts the witness for the four published designs under two hypotheses:

• Q: gravity is a quantum mediator, with the standard Newtonian phase and no first-order-in-\(G\) decoherence of an isolated body ;

• R: the Newtonian field is classical and sourced by a continuous record of the mass density smeared over \(R_0\); consistency of the classical-quantum coupling then forces DP decoherence  at coefficient \(C\ge1\) . The bare DP rate postulate of the companion core paper  leaves \(C\) free at order one; every statement below that holds for \(C\ge1\) also holds for any \(C\) above a design threshold \(C^\ast\sim10^{-3}\).

Both hypotheses are evaluated with the environmental decoherence the designs state: residual gas, black-body photons and the Casimir–Polder interaction.

The results are four. (i) Under Q the partial-transpose witness has a closed form under which-path dephasing, and the Bose et al. design gives a detectable negative value at its stated vacuum (Section 3). (ii) Under R the two-body state is separable at all times for every published design when \(R_0\) is small compared with the bodies, so the concurrence is exactly zero (Section 4). (iii) Only the unscreened Bose geometry closes R’s whole allowed window; we state the requirements for a decisive run and a decision table (Section 5). (iv) The single-arm stage of the same apparatus separates Q from R without the partner mass; the witness adds the separation of Q from classical mediators (Section 6).

A positive witness does not distinguish Q from other quantum theories of gravity: all of them give the same Newtonian phase. The forecast is a test of R and of classical mediators.

2 Model and Designs

2.1 Two bodies, two branches

Two equal spheres \(A\) and \(B\) of mass \(m\) and radius \(a\) are each prepared in \((\ket{L}+\ket{R})/\sqrt2\) and held for a time \(\tau\) with fixed branch positions. With \(r_{ij}\) the distance between branch \(i\) of \(A\) and branch \(j\) of \(B\), the Newtonian energies are \(E_{ij}=-Gm^2/r_{ij}\). Up to local phases, the only nonlocal quantity is \[\begin{equation} \hbar\omega \;=\; \bigl|E_{LL}+E_{RR}-E_{LR}-E_{RL}\bigr| , \qquad \Phi=\omega\tau , \label{eq:omega} \end{equation}\] the entangling phase. We use exact branch distances throughout; no expansion in the arm length is made. The published designs move the branches along trajectories, which changes \(\Phi\) by a factor of order one (Bose et al. quote a phase sum near \(0.5\) rad where the static arms give \(0.314\) rad ); the conclusions below do not depend on that factor.

2.2 Decoherence

Each body suffers which-path dephasing at rate \(\gamma\): every density-matrix element \(\bra{ij}\rho\ket{kl}\) decays as \(e^{-\gamma\tau n}\), where \(n\in\{0,1,2\}\) counts the bodies whose branch differs between bra and ket. This is the convention of the QGEM literature . It is exact for scattering in the saturated regime, where each event resolves the branches.

Under R a second, gravitational channel acts. For a body split into two branches it dephases at \(\Gamma_R = C E_G/\hbar\), with \(E_G=\tfrac12 U(\Delta\rho,\Delta\rho)\), \(U(f,g)=G\iint f(x)g(y)/|x-y|\) evaluated with the \(R_0\)-smeared densities, and for a uniform sphere with \(R_0\ll a\) and branch separation \(\Delta x\ge 2a\), \(E_G=Gm^2(6/(5a)-1/\Delta x)\). The classical mediator is smeared with the same length.

2.3 The designs

Table 1 lists the four designs as published, in diamond (\(\rho=3510\) kg m\(^{-3}\); the dielectric constant \(\varepsilon=5.7\) of Ref. ).

A run is counted as detecting entanglement if \(\langle W\rangle\le-0.0125\): the decoherence-free witness at the minimal detectable phase \(0.05\) rad that Ref.  derives for \(10^4\) runs at five standard deviations.

3 Quantum Gravity: Entanglement at Rate \(\omega/2\)

3.1 The witness under dephasing

The two-qubit state after time \(\tau\) is \(\rho_{ij,kl}=\tfrac14\,p^{\,n}\,e^{-i\Phi(ij-kl)}\) in the labelling \(L=0\), \(R=1\), with \(p=e^{-\gamma\tau}\). Its partial transpose has the eigenvalues \(\tfrac14(1+p^2\pm2p\cos\tfrac{\Phi}{2})\) and \(\tfrac14(1-p^2\pm2p\sin\tfrac{\Phi}{2})\). The expectation of the optimal (partial-transpose) witness is the most negative of them, \[\begin{equation} \langle W\rangle \;=\; \tfrac14\Bigl(1-p^{2}-2p\,\bigl|\sin\tfrac{\Phi}{2}\bigr|\Bigr) , \label{eq:witness} \end{equation}\] in agreement with Eq. (24) of Ref. . For two qubits a negative partial-transpose eigenvalue is necessary and sufficient for entanglement , so \[\begin{equation} \text{entangled at }\tau \iff \sinh(\gamma\tau) < \bigl|\sin\tfrac{\Phi}{2}\bigr| , \qquad \text{entangled at some }t \iff \gamma < \omega_{\mathrm{ent}}\equiv \frac{\omega}{2} . \label{eq:criterion} \end{equation}\] At small \(\gamma\tau\) and \(\Phi\), Eq. \(\eqref{eq:witness}\) reduces to \(\langle W\rangle\approx(\gamma\tau-\Phi/2)/2\), the condition \(\gamma\tau<\Phi/2\) used in Refs. . Equation \(\eqref{eq:criterion}\) is the Q forecast in one line: entanglement grows at the rate \(\omega_{\mathrm{ent}}\) and survives if the environment dephases each body more slowly. The largest tolerable rate at a fixed \(\tau\) is \(\gamma_{\max}=\operatorname{arsinh}(|\sin(\Phi/2)|)/\tau\).

3.2 The environment

Gas collisions. The thermal de Broglie wavelength of residual gas (below a nanometre at \(0.15\) K) is far below every arm, so each collision decoheres. The rate is the collision rate, \[\begin{equation} \gamma_{\mathrm{gas}} \;=\; n\,\pi a^{2}\,\bar v \;=\; 2na^{2}\sqrt{2\pi k_BT/m_{\mathrm{gas}}} , \label{eq:gas} \end{equation}\] the short-wavelength limit derived in Ref.  (\(n=P/k_BT\); air mass \(28.97\) u as in Refs. ). The older expression of Refs.  is larger by \(8\pi/3=8.4\). Residual helium, the gas of a dilution cryostat, raises Eq. \(\eqref{eq:gas}\) by \(\sqrt{28.97/4.00}=2.7\) at fixed pressure.

Black-body photons. At \(0.15\) K the thermal photon wavelength \(\pi^{2/3}\hbar c/k_BT=3.3\) cm exceeds every arm, so scattering, emission and absorption act in the long-wavelength form \(\Lambda\,\Delta x^{2}\) with the localisation rates of Ref. . For the Bose design, scattering is negligible at \(0.15\) K and uses 9% of the decoherence budget at an environment temperature of \(4\) K. Emission depends on the absorptive part of \((\varepsilon-1)/(\varepsilon+2)\); for values between \(10^{-2}\) and \(10^{-4}\) it uses 10% of the budget at an internal temperature of \(0.8\)–\(1.7\) K.

Casimir–Polder. The Casimir–Polder potential between the spheres, \(23\hbar c\,a^{6}f^{2}/(4\pi r^{7})\) with \(f=(\varepsilon-1)/(\varepsilon+2)\) , does not decohere; it entangles, and it does so under both hypotheses. Its ratio to the Newtonian potential, \(23\hbar c f^{2}(3/4\pi\rho)^{2}/(4\pi G r^{6})\), is independent of the mass at fixed density. It is 10% at \(r=157~\mu\)m, the value of Ref. , and 2.3% at the \(200~\mu\)m closest approach of the Bose design, where it adds a phase of \(0.018\) rad to gravity’s \(0.314\) rad.

3.3 Forecast for the published designs

Table 2 applies Eqs. \(\eqref{eq:witness}\)–\(\eqref{eq:gas}\) to the designs of Table 1.

Only the Bose design reaches detection at its own stated environment: \(\langle W\rangle=-0.061\) against \(-0.078\) without decoherence. Its budget \(\gamma_{\max}=0.062~\mathrm{s^{-1}}\) corresponds to a pressure of \(5.1\times10^{-15}\) Pa at \(0.15\) K for gas alone. With the older gas expression the stated vacuum would give \(\gamma=0.10~\mathrm{s^{-1}}\) and no entanglement, so the pressure requirement is uncertain by the factor \(8.4\) between the two expressions. The van de Kamp design entangles but stays below detection. The two Schut designs were built to sit at \(\langle W\rangle\approx0\) for their assumed rates; their margin lies in the trajectory phase, which this static treatment does not resolve.

4 Classical Record-Sourced Gravity: an Exact Null

4.1 The criterion

Under R the two bodies evolve by the DP master equation at coefficient \(C\), which is diagonal in the branch basis, and by the smeared classical Newtonian feedback. For two two-branch bodies the state becomes entangled at some time if and only if \[\begin{equation} C\sqrt{1-r^{2}} \;<\; |r| , \qquad r \;=\; \frac{U(\Delta\rho_A,\Delta\rho_B)}{2\sqrt{E_G^{A}E_G^{B}}} , \label{eq:r-criterion} \end{equation}\] with all quantities smeared over \(R_0\). At \(C=1\) this is the criterion of Trillo and Navascués , whose critical distances \(0.850872\,\sigma\) (collinear) and \(2.21093\,\sigma\) (transverse) it reproduces; Eq. \(\eqref{eq:r-criterion}\) extends it to any \(C\). By the Cauchy–Schwarz inequality for the positive Newton form, \(|r|\le1\). For uniform spheres with \(R_0\ll a\) that cannot overlap, the touching configuration with arms transverse to the axis gives \(|r|\to5/12\) as the arms grow, and a numerical scan of other geometries stays below it; no two such spheres then entangle for \(C>5/\sqrt{119}=0.458\).

4.2 The published designs

For \(R_0\ll a\) the threshold \(C^\ast=|r|/\sqrt{1-r^{2}}\) is \(7.3\times10^{-4}\) (Bose et al.), \(2.3\times10^{-3}\) (van de Kamp et al.), \(5.1\times10^{-4}\) (Schut et al. 2023) and \(2.0\times10^{-4}\) (Schut et al. 2024). At \(C\ge1\), and at any order-one \(C\), the partial transpose of the full \(4\times4\) state stays positive at every time. For two qubits that means separable : the concurrence is exactly zero. The R prediction is a null, not a suppressed signal.

A larger smearing length can lift the null, because it weakens the self-decoherence of each body more than the cross term. Equation \(\eqref{eq:r-criterion}\) at \(C=1\) allows entanglement only above the lengths \(R_0^{\ast}\) of Table 3; a larger \(C\) raises them. \(R_0^{\ast}\) tracks the centre separation \(D\): it is \(1.17\,D\) for collinear arms and \(0.65\,D\) for parallel arms. Because R’s mean force is fed back from the smeared density, tests of the inverse-square law down to \(52~\mu\)m  bound \(R_0\); we take \(R_0\le4\times10^{-5}\) m, and \(10^{-4}\) m as a conservative value.

The Bose geometry gives an R null for every \(R_0\) up to \(0.52\) mm, beyond both bounds. The van de Kamp and Schut 2023 geometries close the window at the \(40~\mu\)m bound but not at \(10^{-4}\) m. The Schut 2024 geometry leaves \(R_0\) between \(14\) and \(40~\mu\)m open: there a positive witness would not exclude R.

4.3 The single-body channel

R also predicts that each body loses its own coherence, at the rate \(\Gamma_R\) of Section 2. In the Bose run the single-body exponent is \(215\,C\) for \(R_0\ll a\) and \(1.6\,C\) at \(R_0=40~\mu\)m, against \(0.031\) from gas collisions under Q. The residual Casimir–Polder phase of \(0.018\) rad, which would entangle under R as well, is erased by this channel at every \(R_0\le40~\mu\)m and lies below detection on its own.

5 The Decisive Run and the Decision Table

5.1 Requirements

A run decides between Q and R if it gives a detectable witness under Q, keeps the Casimir–Polder phase below gravity’s, and gives an R null across R’s allowed window. Without a screening plate the second condition fixes the closest approach at \(d_c\ge157~\mu\)m for any mass; the third then holds automatically, since \(R_0^{\ast}\approx1.17\,D\) for collinear arms and \(D>157~\mu\)m. Table 4 gives the requirements for diamond in the linear geometry with \(d_c=157~\mu\)m and \(\Delta x=250~\mu\)m.

At fixed \(\gamma\) the allowed pressure grows as \(\sqrt{T}\): \(1.7\times10^{-14}\) Pa at \(1\) K for the \(10^{-14}\) kg, \(2.5\) s run. The threshold mass is \(5.2\times10^{-15}\) kg at \(\tau=1\) s, and \(4.8\times10^{-15}\) kg for a \(100~\mu\)m arm at \(2.5\) s. The environment temperature must stay below about \(4\) K and the internal temperature below about \(1\) K (Section 3). A screening plate can replace the \(157~\mu\)m condition, but not the third: R’s window closes only if \(D\gtrsim34~\mu\)m (collinear) or \(62~\mu\)m (parallel) at the \(40~\mu\)m bound, and \(86\) or \(155~\mu\)m at \(10^{-4}\) m.

Among the published designs only the Bose geometry meets all three conditions. Its decoherence budget at its stated temperature is a pressure of \(5.1\times10^{-15}\) Pa.

5.2 Decision table

The same apparatus measures two quantities: the single-arm interference contrast of one particle, and the two-body witness.

• Contrast and witness both present. Q. R is excluded for every \(C\ge1\) and every \(R_0\) up to \(0.52\) mm; classical mediators acting by local operations are excluded.

• Contrast present, witness absent at the predicted sensitivity. The contrast excludes R (\(C<0.019\) at \(R_0=40~\mu\)m for contrast \(0.97\)), and the missing witness contradicts Q’s Newtonian phase. Neither hypothesis survives.

• Contrast absent with the environment certified, by the scaling of the loss with pressure and temperature. R, with \(C\) measured from the loss and its \(m^{5/3}\) scaling at fixed density.

6 Discussion and Conclusions

6.1 What the witness adds

The decision between Q and R is already made by the single-arm contrast (Table 5): R predicts that a \(10^{-14}\) kg body split over \(250~\mu\)m loses at least 80% of its contrast within the run (\(\Gamma_R\tau\ge1.6\) at \(C=1\)) for every \(R_0\le40~\mu\)m, and Q predicts that only the environment acts. The partner mass is not needed for that call. The witness adds the separation of Q from classical mediators that do not decohere a single body, and the certification that the Newtonian phase is carried coherently. Both experiments need the same particle, arm and vacuum; the single arm needs half the apparatus.

6.2 What the forecast does not claim

• A positive witness does not select Q among quantum theories of gravity: all of them give the Newtonian phase of Eq. \(\eqref{eq:omega}\).

• The forecast does not test the rate postulate of Ref.  by itself; it tests R, which fixes \(C\ge1\) and a classical mediator. For a bare DP rate with an unspecified mediator the null of Section 4 still holds for \(C>C^\ast\), which includes every order-one value.

• Static arms change the phases by a factor of order one; spin dephasing, magnetic-gradient noise and patch potentials are not modelled and add to \(\gamma\), which Table 4 states as a total budget.

6.3 Distance to experiment

The decisive run requires a \(250~\mu\)m superposition of a \(10^{-14}\) kg body for about a second. The largest coherent delocalisation of a levitated solid reported so far is \(73\) pm for a \(1.2\) fg silica sphere : about \(10^{4}\) short in mass and \(3\times10^{6}\) in length. The BMV witness is therefore not a discriminator between Q and R within the present decade.

6.4 Conclusions

1. Under quantum gravity with which-path dephasing at rate \(\gamma\), the partial-transpose witness is \(\tfrac14(1-p^{2}-2p|\sin(\Phi/2)|)\) and entanglement appears iff \(\gamma<\omega/2\). The Bose et al. design gives \(\langle W\rangle=-0.061\) at its stated vacuum.

2. Under classical record-sourced gravity with DP decoherence at \(C\ge1\) the two-body state is separable at all times in every published design for \(R_0\ll a\), and in the Bose geometry for \(R_0\) up to \(0.52\) mm: the concurrence is exactly zero.

3. A decisive run needs \(d_c\ge157~\mu\)m, \(m\ge3.3\times10^{-15}\) kg, an arm of \(100\)–\(250~\mu\)m, \(\tau\sim1\)–\(2.5\) s and a pressure below \(6.6\times10^{-15}\) Pa at \(0.15\) K; only the unscreened Bose geometry meets these conditions among the published designs.

4. The single-arm contrast of the same particle separates Q from R without the witness.

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