Two masses, each in a superposition of positions, become entangled through their gravitational interaction if gravity is a quantum channel; the detection of that entanglement has been proposed as a witness of the quantum nature of gravity [1, 2]. The witness needs a classical benchmark. The Diósi–Penrose model [3, 4, 5], in the form of Tilloy and Diósi [6, 7], is one: a classical Newtonian field sourced by continuously monitored, \(\sigma\)-smeared mass densities, with a decoherence that the positivity of the dynamics forces [8]. Trillo and Navascués showed that this model does entangle two masses, in the same geometries in which the quantum prediction is made [9]. For two branches per mass they found critical distances, \(0.850872\,\sigma\) for collinear superpositions of vanishing size and \(2.21093\,\sigma\) for transversal superpositions of diverging size, and asked: “Is there a way to observe GIE beyond \(d_c\)?” Later work quotes \(d_c\approx0.85\,\sigma\) as the separation below which the model entangles [10, 11]; that statement is correct for small packets with two branches, and it is the general reading of it that this paper corrects. Wang, Pedernales and Plenio, in a study of local error correction, gave a separability lemma for one pair of coupled operators, found the Diósi–Penrose dynamics of their two-level codes separable outside a small region of \(d/\sigma\), and left the exact classification as “beyond our scope” [12].
This paper gives that classification for position branches and draws the consequences. The result has two parts.
The criterion (Theorem 2.2). With no kinetic term, two rigid bodies in a product state on finite sets of position branches become entangled if and only if \[\begin{equation} \rho_{\max}(S_A,S_B)\;>\;\frac{C}{\sqrt{1+C^2}}, \label{eq:headline-criterion} \end{equation}\] where \(\rho_{\max}\) is the largest canonical correlation [13], in the Coulomb energy of the smeared bodies, between neutral charge patterns on the two branch sets, and \(C\geq1\) is the noise strength (\(C=1\) in [9]). Below the threshold the evolution is a separable operation for all time. The criterion depends on the branch sets, \(\sigma\), the body shape and \(C\); not on the masses, the amplitudes or the time.
The size (Sections 4 and 5). With many branches the threshold is exceeded at every separation, so there is no critical distance. But the rate at which entanglement is produced is a small fixed fraction of the rate at which each body dephases under the same dynamics, \[\begin{equation} \frac{\Gamma_{\rm ent}}{\Gamma_{\rm sat}}\;=\;\frac{\sqrt{\pi}\,\kappa}{C}\,\frac{\sigma_{\rm eff}}{d} \;\leq\;0.12\,\frac{\sigma_{\rm eff}}{d}\qquad(C=1,\ \text{slab-separated supports},\ \sigma\ll d), \label{eq:headline-ratio} \end{equation}\] with \(\kappa\) a dimensionless rate constant, \(\Gamma_{\rm sat}= C\,Gm^2/(\sqrt{\pi}\,\hbar\,\sigma_{\rm eff})\) the saturated Diósi–Penrose dephasing rate, and \(\sigma_{\rm eff}=\sigma\) for point masses. The negativity peaks at about \(\Gamma_{\rm ent}/(2e\Gamma_{\rm sat})\), half a dephasing time after preparation. Section 5 applies both parts to the design of Bose et al. [1]; Section 6 lists what is not controlled.
The Tilloy–Diósi dynamics with smeared force and noise, in the notation of [9] with a noise strength \(C\), is \[\begin{equation} \dot\rho=-\frac{i}{\hbar}[H,\rho]+\frac{G}{2\hbar}\int\!\!\int\frac{d^3r\,d^3s}{|\mathbf r-\mathbf s|} \Big(-C\,\big[\mu_\sigma(\mathbf r),[\mu_\sigma(\mathbf s),\rho]\big] +i\,\big[\mu_\sigma(\mathbf r)\mu_\sigma(\mathbf s),\rho\big]\Big), \label{eq:master} \end{equation}\] with \(\mu_\sigma\) the mass density smeared by a normalised Gaussian of standard deviation \(\sigma\). The second term is the Newtonian interaction of the smeared densities, the first the monitoring noise. In measurement-and-feedback realisations \(C\geq1\), and \(C=1\) is the least-decoherence point [6, 7]; \(C=1\) is the case of [9].
Two rigid bodies \(A\), \(B\) with masses \(m_A\), \(m_B\) carry their centres as quantum degrees of freedom. Let \(g_a\) be the \(\sigma\)-smeared unit-mass density of body \(A\) centred at \(a\), and \(g_b\) that of \(B\). On charge densities use the Coulomb form \(\langle u,v\rangle=\int\!\!\int u(x)v(y)/|x-y|\) and put \[\begin{equation} f_A(a,a')=\langle g_a,g_{a'}\rangle,\qquad f_B(b,b')=\langle g_b,g_{b'}\rangle,\qquad F(a,b)=\langle g_a,g_b\rangle. \label{eq:kernels} \end{equation}\] For point masses all three equal \(f(r)=\operatorname{erf}(r/2\sigma)/r\), with \(f(0)=f_0=1/(\sqrt{\pi}\,\sigma)\). With \(H=0\), Eq. \(\eqref{eq:master}\) acts on position matrix elements multiplicatively, \[\begin{align} \rho_t(ab,a'b')&=\rho_0(ab,a'b')\,\exp\!\Big[\frac{Gt}{\hbar}\,\Lambda(ab,a'b')\Big],\label{eq:solution}\\ \Lambda&=i\,m_Am_B\big[F(a,b)-F(a',b')\big]-\frac{C}{2}\,\big\|m_A(g_a-g_{a'})+m_B(g_b-g_{b'})\big\|^2 . \label{eq:Lambda} \end{align}\] The norm is the Coulomb norm. Expanded, the real part is \(-C\,[m_A^2D_A+m_B^2D_B+m_Am_B\,\delta]\) with \(D_A=f_A(a,a)-f_A(a,a')\) and \(\delta=F(a,b)+F(a',b')-F(a,b')-F(a',b)\). The local terms \(D_A\), \(D_B\) are the Diósi–Penrose dephasing of each body; \(\delta\) is the correlated part of the noise.
A state is a branch state on finite sets \(S_A\), \(S_B\) of centre positions if it is supported on \(S_A\times S_B\); for a product state \(\psi_A\otimes\psi_B\) write \(p_a=|\psi_A(a)|^2\), \(r_b=|\psi_B(b)|^2\). The branches are position eigenstates (Section 6 discusses wavepackets).
Definition 2.1. Let \(V_A=\mathrm{span}\{g_a-g_{a'}:a,a'\in S_A\}\) and \(V_B\) likewise: the neutral charge patterns on the branch sets. Their largest Coulomb canonical correlation is \[\begin{equation} \rho_{\max}(S_A,S_B)=\sup\Big\{\frac{|\langle u,v\rangle|}{\|u\|\,\|v\|}:\ u\in V_A,\ v\in V_B\Big\}, \qquad s_*(C)=\frac{C}{\sqrt{1+C^2}} . \label{eq:rhomax} \end{equation}\]
\(\rho_{\max}\) is the top singular value of \(f_A^{-1/2}Ff_B^{-1/2}\) restricted to zero-sum vectors; it does not depend on the masses. \(s_*(1)=1/\sqrt2=0.7071\).
Theorem 2.2. Let \(H=0\) and \(C>0\), and let \(S_A\), \(S_B\) be finite sets of distinct branches.
If \(\rho_{\max}>s_*\), every pure product state with all amplitudes non-zero has a partial transpose with a negative eigenvalue for all sufficiently small \(t>0\). That eigenvalue leaves zero at the rate \[\begin{equation} \dot\lambda=\frac{G}{\hbar}\min_{q_1,q_2} \frac{C m_A^2\langle q_1,q_1\rangle+C m_B^2\langle q_2,q_2\rangle-2m_Am_B\,\mathrm{Re}\big[(C+i)\langle q_1,q_2\rangle\big]} {\sum_a|q_1(a)|^2/p_a+\sum_b|q_2(b)|^2/r_b}, \label{eq:rate} \end{equation}\] the minimum over neutral complex patterns \(q_1\) on \(S_A\) and \(q_2\) on \(S_B\), with \(\langle q_1,q_2\rangle=\sum_{ab}\bar q_1(a)q_2(b)F(a,b)\) and \(\langle q_1,q_1\rangle=\sum\bar q_1(a)q_1(a')f_A(a,a')\).
If \(\rho_{\max}\leq s_*\), the evolution restricted to \(S_A\times S_B\) is a separable operation for every \(t\geq0\): a mixture of products of local completely positive maps. Every separable initial state stays separable, also when local channels acting within the branch spaces are interleaved.
Hence the model entangles some product state on \(S_A\times S_B\) at some time if and only if it entangles every full-support pure product state at first order, if and only if \(\rho_{\max}>s_*\).
Proof sketch. (a) Partial transposition exchanges \(a\leftrightarrow a'\), and \(\rho_0^{T_1}\) is a rank-one projector. For a vector \(\chi\) orthogonal to it put \(c=\tilde\Psi^*\chi\) entrywise, \(\tilde\Psi=\bar\psi_A\otimes\psi_B\), so that \(\sum c=0\). At first order in \(t\) every term of \(\Lambda\) that depends on \((a,b)\) alone or on \((a',b')\) alone drops out of \(\langle\chi|\rho_t^{T_1}|\chi\rangle\), and with the marginals \(q_1(a)=\sum_bc(ab)\), \(q_2(b)=\sum_ac(ab)\) what remains is the numerator of Eq. \(\eqref{eq:rate}\). Every pair of neutral patterns arises, through \(c=q_1\otimes r+p\otimes q_2\), which also minimises \(\|\chi\|^2\) at given marginals and gives the denominator. Optimising the phase and the scale of \(q_2\), the numerator is negative for some pair if and only if \(|\langle q_1,q_2\rangle|^2>s_*^2\langle q_1,q_1\rangle\langle q_2,q_2\rangle\), that is, if and only if \(\rho_{\max}>s_*\); the state is then entangled by the Peres criterion [14].
(b) Choose bases of \(V_A\) and \(V_B\) that diagonalise the cross form (a singular value decomposition with singular values \(s_i\leq\rho_{\max}\)) and let \(x_i(a)\), \(y_i(b)\) be the coordinates of the branches in them. Since one kernel \(F\) enters both the force and the correlated noise, and the local dephasing is the Gram form used for the whitening, \[\begin{equation} \Lambda=\sum_i\Lambda_i+(\text{local phases}),\qquad \Lambda_i=is_i(x_iy_i-x_i'y_i')-\tfrac{C}{2}(x_i-x_i')^2-\tfrac{C}{2}(y_i-y_i')^2-Cs_i(x_i-x_i')(y_i-y_i'), \label{eq:pairs} \end{equation}\] with no terms coupling different \(i\) (masses absorbed into \(g\)). Each \(\Lambda_i\) is reproduced by random local phase-and-amplitude multipliers \(e^{x_i(\eta_i+i\zeta_i)}\), \(e^{y_i(\eta_i'+i\zeta_i')}\) with Gaussian latent variables, up to a local Schur multiplier with exponent \(t\,(C-s_i\sqrt{1+C^2})\,x_ix_i'\). When \(s_i\sqrt{1+C^2}\leq C\) for every \(i\) that multiplier is positive semidefinite, hence completely positive, and \(\rho_t\) is an average of products of local completely positive maps applied to \(\rho_0\). Appendix 7 gives the covariance and the bookkeeping. (c) follows from (a) and (b). ◻
Remark 2.1. The proof uses only that one real cross kernel enters both the Hamiltonian and the correlated noise and that the local dephasing is the Gram form. Theorem 2.2 therefore holds for any real cross kernel \(\bar F\) with \(\rho_{\max}\) replaced by the top singular value of \(f_A^{-1/2}\bar Ff_B^{-1/2}\) on neutral vectors, provided that value is at most one (otherwise the noise is not positive). Theorem 4.2 below uses this for \(\bar F=\theta F\).
Remark 2.2. Each \(\Lambda_i\) in Eq. \(\eqref{eq:pairs}\) is the generator of Lemma 3 of [12] with \(O_A=\sum_ax_i(a)|a\rangle\langle a|\), \(O_B=\sum_by_i(b)|b\rangle\langle b|\), coupling \(\lambda=-s_i\), local rates \(\Gamma_A=\Gamma_B=C\) and cross rate \(\Gamma_{AB}=-Cs_i\) (in units \(Gm^2/\hbar\sigma\)). Their lemma makes \(e^{t\Lambda_i}\) separable when \(\lambda^2+\Gamma_{AB}^2\leq\Gamma_A\Gamma_B\), which is \(s_i\sqrt{1+C^2}\leq C\). Part (b) therefore also follows from their lemma applied pair by pair, for one reason: the canonical decomposition leaves no cross terms between pairs, so \(e^{t\Lambda}\) is a product of separable channels up to a local unitary. That the \(\Lambda_i\) commute is automatic for position-diagonal generators and is not the reason; for several coupled operators a pairwise condition does not in general give a joint one (cf. their Theorem 1). With an unsmeared force, force and noise have different kernels, no single decomposition diagonalises both, and the pairwise argument fails for three or more branches. Necessity for any number of branches is part (a).
Two features of the threshold are worth stating. The noise helps: dephasing alone would give the threshold \(1\), never reached; the Hamiltonian supplies the \(i\) and the correlated noise the \(C\) in \(C+i\), and \(|C+i|=\sqrt{1+C^2}\) lowers the threshold to \(s_*<1\). And mode by mode the generator is never positive: for one Fourier mode the two-by-two form has determinant \(C^2-(1+C^2)=-1\), so separability comes only from the two bodies occupying different regions.
Remark 2.3. Potentials of smeared charges are real-analytic, so the smeared point charges of any open set span the whole Coulomb space: \(\rho_{\max}=1\) for any two supports with interior. Every pair of square-integrable wavefunctions is therefore entangled at once, at every separation (Theorem 7.1 in Appendix 7 gives the entangling direction in \(L^2\)). The “if and only if” of Theorem 2.2 is a statement about position-eigenstate branches, the idealisation also used in [9, 12]. The physical content lies in the rate and the size of the entanglement, Section 4.
For \(S_A=\{a_1,a_2\}\), \(S_B=\{b_1,b_2\}\) each \(V\) is one-dimensional and Eq. \(\eqref{eq:rhomax}\) is \[\begin{equation} s=\frac{|f(a_1b_1)+f(a_2b_2)-f(a_1b_2)-f(a_2b_1)|}{2\sqrt{D(L_A)D(L_B)}},\qquad D(L)=f_0-f(L), \label{eq:twobranch} \end{equation}\] with \(L_A=|a_1-a_2|\), \(L_B=|b_1-b_2|\) and \(f(xy)=f(|x-y|)\) (point masses). Four limiting families have closed forms (Table 1, Fig. 1); each family increases monotonically towards its limit, which is not attained. With \(\kappa_0=-f''(0)=1/(6\sqrt{\pi}\,\sigma^3)\): superpositions of length \(L\to0\) along the line joining the bodies give \(|f''(d)|/\kappa_0\); transversal superpositions give \(|f'(d)/d|/\kappa_0\) for \(L\to0\) and \(f(d)/f_0=\sqrt{\pi}\,\sigma\operatorname{erf}(d/2\sigma)/d\) for \(L\to\infty\); and a vanishing superposition of \(A\) midway between two branches of \(B\), each at distance \(d\) (“straddle”), gives \(\sqrt2\,|f'(d)|/\sqrt{\kappa_0(f_0-f(2d))}\). Setting each equal to \(s_*\) reproduces the two distances of [9] and adds two. The straddle limit exceeds the transversal one for \(d<3.699\,\sigma\), that is for every \(C>0.5396\).
| family | limit of \(s\) | \(C=1\) | \(C=1.25\) | \(C=2\) | \(C=5\) |
|---|---|---|---|---|---|
| collinear, \(L\to0\) [9] | \(|f''(d)|/\kappa_0\) | 0.850872 | |||
| transversal, \(L\to0\) | \(|f'(d)/d|/\kappa_0\) | 1.547134 | |||
| transversal, \(L\to\infty\) [9] | \(f(d)/f_0\) | 2.210928 | 1.8203 | 1.1845 | 0.4870 |
| straddle | \(\sqrt2|f'(d)|/\sqrt{\kappa_0(f_0-f(2d))}\) | 2.738073 | 2.4425 | 1.9106 | 1.1832 |
Most designs give both masses the same superposition, translated; the design of [1] is one. For that class the transversal limit is the supremum.
Theorem 3.1. Let \(S_A=\{a,a+w\}\) and \(S_B=\{a+r,a+r+w\}\) for arbitrary vectors \(w\), \(r\), and let \(d\) be the closest approach. Then \[\begin{equation} s\leq\max\big(f(d)/f_0,\;c_*\big),\qquad c_*=\Lambda_+L_{1/2}^2/f_0=0.634687, \label{eq:parallel} \end{equation}\] where \(\Lambda_+=\max f''=0.311326\,\kappa_0\) (at \(r=3.2273\,\sigma\)) and \(f(L_{1/2})=f_0/2\), \(L_{1/2}=3.497418\,\sigma\). For every \(C\geq0.8213\), in particular every \(C\geq1\), no parallel two-branch state with closest approach \(d\geq d_*(C)\) is entangled, where \(f(d_*)/f_0=s_*(C)\) (\(d_*=2.210928\,\sigma\) at \(C=1\)); for every \(d<d_*(C)\) some are.
The proof (Appendix 8) uses that \(h(x)=f(\sqrt x)\) is a mixture of decaying exponentials, hence convex and log-convex. For arbitrary two-branch geometries we have only computer-assisted ceilings.
Proposition 3.2. For every \(C\geq1\) no two-branch state is entangled by the model at slab separation \(d\geq2.75\,\sigma\) (the branches of \(A\) and \(B\) on opposite sides of a plane, at normal distance \(\geq d\)), nor at closest approach \(d\geq3.78\,\sigma\).
The bound on \(s\) combines four rigorous inequalities (Appendix 8); its final maximisation over the two splittings is a floating-point evaluation of cell bounds on a geometric grid, without interval arithmetic. It gives \(0.70456\) at \(2.75\,\sigma\) (slab) and \(0.70457\) at \(3.78\,\sigma\) (closest approach), \(0.36\%\) below \(1/\sqrt2\), and exceeds \(1/\sqrt2\) at \(2.735\,\sigma\) (\(0.70875\)) and \(3.765\,\sigma\) (\(0.71019\)). A numerical search over all two-branch geometries (differential evolution followed by Nelder–Mead, over several parametrisations) found nothing above \(f(d)/f_0\) at slab separation, or above \(\max(f(d)/f_0,\text{straddle limit})\) at closest approach; we have not proven these to be the suprema. So at \(C=1\) the largest two-branch distance lies in \([2.2109,2.75]\,\sigma\) (slab) and \([2.7381,3.78]\,\sigma\) (closest approach), and equals \(2.2109\,\sigma\) within the parallel class.
Theorem 3.3. For every \(d>0\) and every \(C<\infty\) there are finite branch sets at slab separation \(d\) with \(\rho_{\max}>s_*\). For uniform coaxial discs of radius \(a\) at gap \(d\) (lattices of point branches), each body carrying in addition a far copy of its disc, \[\begin{equation} \rho_{\max}\geq\operatorname{erf}(d/2\sigma)\Big(1-\frac{3\pi d}{8a}\Big), \label{eq:discbound} \end{equation}\] so the model entangles such states once \(a/d>(3\pi/8)/\big(1-s_*/\operatorname{erf}(d/2\sigma)\big)\), which is \(4.022\) at \(C=1\) and \(d\gg\sigma\).
The condition is on the extent of the coherent superposition in units of the gap, not on \(d/\sigma\). The far copy makes the disc pattern neutral without changing the correlation; it means that the coherent extent of the state is really the distance to the copy. The sharp version of this extent law is in Section 4.3. Since \(\rho_{\max}=1\) for every pair of open supports (Remark 2.3), the absence of a critical distance is not by itself a physical prediction. What matters is how much entanglement the model can produce, and how fast.
For equal masses write the rate \(\eqref{eq:rate}\) as \(\dot\lambda=-\kappa\,Gm^2/(\hbar d)\), with \(d\) the separation and \(\kappa\geq0\) dimensionless. Eq. \(\eqref{eq:rate}\) is also the initial rate when a local Hamiltonian is present [9].
Theorem 4.1. Let \(\theta=\min(1,s_*/\rho_{\max})\).
For every product state with closest approach \(d\) and every \(\sigma\), \(\kappa\leq(1-\theta)\sqrt{1+C^2}\,d f(d)\leq\sqrt{1+C^2}-C\), which is \(0.4142\) at \(C=1\).
For supports separated by a slab of width \(d\), with the Coulomb kernel (\(\sigma\to0\)), \[\begin{equation} \kappa\leq B(C)=\sqrt{1+C^2}-C-C\ln\frac{\sqrt{1+C^2}}{C}, \label{eq:BC} \end{equation}\] \(B(1)=0.067640\), \(B(C)\to1/(8C^3)\) for \(C\to\infty\). For planar supports at finite \(\sigma\) the corresponding bound is larger, \(0.104\) at \(d=3\,\sigma\) and \(C=1\), so \(\eqref{eq:BC}\) is a statement for \(\sigma\ll d\).
Explicit states bound the best \(\kappa\) from below. Give each mass an equal superposition of a planar sheet and a far copy of it, the two sheets facing each other at gap \(d\), in the formal limits \(\sigma\to0\) and zero thickness. A Gaussian sheet of rms width \(b\) has, for its natural witness, \[\begin{equation} \kappa_G=\max_{y}\ \frac{\sqrt\pi}{2}\,y\big[\sqrt{1+C^2}\,e^{y^2}\operatorname{erfc}(y)-C\big],\qquad y=\frac{d}{2b}, \label{eq:kappaG} \end{equation}\] \(\kappa_G=0.027108\) at \(b=3.15\,d\) (\(C=1\)); a uniform disc gives \(0.0264\) at \(a=6.12\,d\). So at \(C=1\) and \(\sigma\ll d\) the best rate constant of slab-separated states lies between \(0.027\) and \(0.068\). Both explicit values are proven formulas for one witness each, in limits we do not control; at finite \(d/\sigma\) smearing raises them by corrections of first order in \(\sigma/d\).
In the same dynamics the coherence between two branches \(a\), \(a'\) of one body decays at the Diósi–Penrose rate \(C\,Gm^2[f(0)-f(|a-a'|)]/\hbar\). Because the body is rigid and only its centre is delocalised, this saturates for \(|a-a'|\) larger than a few \(\sigma_{\rm eff}\): \[\begin{equation} \Gamma_{\rm sat}=\frac{C\,Gm^2K_0}{\hbar},\qquad K_0=\frac{1}{\sqrt\pi\,\sigma_{\rm eff}}, \label{eq:Gsat} \end{equation}\] with \(\sigma_{\rm eff}=\sigma\) for point masses and \(K_0=6/(5R)\), \(\sigma_{\rm eff}=5R/(6\sqrt\pi)=0.470\,R\), for uniform spheres of radius \(R\gg\sigma\). A wide superposition does not dephase faster per pair; every pair of sites further apart than \(\sigma_{\rm eff}\) already dephases at \(\Gamma_{\rm sat}\). For extended bodies Theorem 2.2 and the bounds hold verbatim with \(f\) replaced by the Coulomb energy \(K(r)\) of two smeared bodies, which is \(1/r\) beyond contact; at large separation only the self-term \(K_0\) changes. With \(\Gamma_{\rm ent}=|\dot\lambda|=\kappa\,Gm^2/(\hbar d)\), \[\begin{equation} \mathcal{R}\equiv\frac{\Gamma_{\rm ent}}{\Gamma_{\rm sat}}=\frac{\kappa}{CK_0d}=\frac{\sqrt\pi\,\kappa}{C}\,\frac{\sigma_{\rm eff}}{d}. \label{eq:ratio} \end{equation}\] This is an identity; its bounds are those of Theorem 4.1 (Table 2, Fig. 2). At large \(C\), \(\mathcal{R}\leq(\sqrt\pi/8C^4)\,\sigma_{\rm eff}/d\) for slab-separated supports. At \(C=1\) each body’s coherence is lost at least \(8.3\,d/\sigma_{\rm eff}\) times faster than entanglement is generated (slab-separated, \(\sigma\ll d\)), and \(20.8\,d/\sigma_{\rm eff}\) times faster for the best state known. As a function of the extent \(\mathcal{R}\) vanishes below a threshold (\(b/d\leq1.465\) for Gaussian sheets at \(C=1\)), peaks near an extent of a few gaps, and falls off like gap over extent beyond: a wider sheet is not better.
| \(C\) | Gaussian sheets | slab-separated, ceiling | any product state, ceiling |
|---|---|---|---|
| 1 | 0.0481 | 0.120 | 0.734 |
| 1.25 | 0.0235 | 0.0590 | 0.497 |
| 2 | 0.00453 | 0.0115 | 0.209 |
| 5 | 0.000135 | 0.00034 | 0.035 |
The dephasing is already inside the criterion. The first term of the numerator of \(\eqref{eq:rate}\), \(C\) times the Coulomb self-energy of the witness pattern, is the Diósi–Penrose dephasing rate of the collective coherence that the witness reads. For a smooth pattern over \(n\) point-like branches it consists of a collective part and the point self-energies \(f_0\sum|q(a)|^2\), which are diluted as \(1/n\). To leading order, the criterion \(\rho_{\max}>s_*\) then reads \(n\,\mathcal{R}_{\rm coll}>1\), with \(\mathcal{R}_{\rm coll}\) the ratio of the collective rate to \(\Gamma_{\rm sat}\): the saturated dephasing enters divided by the number of branches.
For coaxial discs of radius \(a\) at gap \(d\gg\sigma\), each with a far copy, the criterion becomes scale-free (formal limit \(\sigma\to0\), zero thickness). The uniform pattern is a witness if and only if \[\begin{equation} \Phi(a/d)>s_*,\qquad \Phi(x)=\frac{3\pi}{16}\int_0^\infty\Big(\frac{2J_1(u)}{u}\Big)^2e^{-u/x}\,du . \label{eq:Phi} \end{equation}\] For all axisymmetric patterns, writing the Hankel transform of the charge as an Abel transform \(\hat q(k)=\int_0^1g(t)\cos(kt)\,dt\) turns the canonical correlation into the top eigenvalue of Love’s operator for the circular plate condenser [16], \[\begin{equation} (T_\delta g)(t)=\frac1\pi\int_{-1}^{1}\frac{\delta\,g(s)}{\delta^2+(t-s)^2}\,ds,\qquad\delta=d/a, \label{eq:love} \end{equation}\] and the criterion is \(\lambda_{\max}(T_{d/a})>s_*\) (Appendix 8). Table 3 gives the thresholds; the eigenvalues are computed by Nyström discretisation and are converged. At \(C=1\) the coherent diameter must exceed \(5.54\) gaps (\(5.66\) for uniform amplitude); at large \(C\), \(1-\Phi\simeq(3\pi/8)\,d/a\) gives \(a/d\simeq(3\pi/4)C^2=2.356\,C^2\) for the uniform pattern. Without the far copy, the best neutral pattern found on a single disc is dipolar, the odd block of Love’s operator: it needs \(a/d>7.390\), a coherent diameter of \(14.8\) gaps, against \(11.34\) (\(22.7\) gaps) for the best axisymmetric neutral pattern. In a lattice with spacing larger than \(\sigma\), reaching the threshold needs at least about \(15\,d/\sigma\) point-like branches per mass (numerically, \(15.2\,d/\sigma\) at \(d=12\,\sigma\) and \(15.6\,d/\sigma\) at \(d=24\,\sigma\), hexagonal patches with a far copy at \(C=1\)).
| \(C\) | uniform pattern | optimal axisymmetric pattern |
|---|---|---|
| 1 | 2.8316 | 2.7706 |
| 1.25 | 4.1316 | 4.0642 |
| 2 | 9.7539 | 9.6453 |
| 5 | 58.810 | 58.041 |
Below, the negativity is \(|\lambda_{\min}(\rho_t^{T_1})|\), the quantity plotted in [9].
Theorem 4.2. For \(H=0\), equal masses, any product state with weights \(p\), \(r\) on finite branch sets, \(\theta\) as in Theorem 4.1, and all \(t'=Gm^2t/(\hbar\sigma)\geq0\) (lengths in units of \(\sigma\)), \[\begin{equation} |\lambda_{\min}(\rho_t^{T_1})|\leq(1-\theta)\,t'\Big\{\sum p_ap_{a'}r_br_{b'}\,(C^2\delta^2+\epsilon^2)\, e^{-2t'C\,(D_A+D_B+\min(\theta\delta,\delta))}\Big\}^{1/2}, \label{eq:negbound} \end{equation}\] the sum over all \(a,a',b,b'\), with \(\epsilon=F(a,b)-F(a',b')\). If the branches of each mass are at least \(\ell\) apart, \(D_\ell=f_0-f(\ell)>f(d)\) and \(C\geq1/\sqrt3\), then \(\sup_t|\lambda_{\min}|\leq(1-\theta)(f(d)/e)\big[1/(D_\ell-f(d))+\sqrt{\sum p^2+\sum r^2}/(CD_\ell)\big]\); for \(\ell,d\gg\sigma\) and many branches this is at most \((\sqrt\pi/e)(1-s_*)\,\sigma/d\): \(0.191\,\sigma/d\) at \(C=1\), \(0.069\,\sigma/d\) at \(C=2\).
The proof (Appendix 8) compares with the model whose cross kernel is \(\theta F\), which is separable by Remark 2.1. The clean \(\sigma/d\) form needs branches more than a few \(\sigma\) apart; for states with structure below \(\sigma\) the sum \(\eqref{eq:negbound}\) has to be evaluated as it stands, and a \(\sigma/d\) bound for all product states is not proven.
A leading-order law. Expanding the exact matrix elements to first order in the cross kernel, with the local dephasing kept exactly, gives for \(n\) branches per mass with uniform weights \(n\lambda_{\min}\simeq e^{-2\tau}[e^\tau-1-\xi\tau]\) up to terms of relative order \(1/n\), \(\tau=\Gamma_{\rm sat}t\), with \(\xi=n\mathcal{R}_{\rm coll}\) the quantity whose exceeding one is the criterion. The factor \(e^{-2\tau}\) is the survival of one coherence on each body, \(e^\tau-1\) the mixedness the dephasing creates in the witness subspace, \(-\xi\tau\) the entangling term. For \(\xi\gg1\), \[\begin{equation} \lambda_{\min}(t)\simeq-\Gamma_{\rm ent}\,t\,e^{-2\Gamma_{\rm sat}t},\qquad \max_t|\lambda_{\min}|\simeq\frac{\mathcal{R}}{2e}=\frac{\sqrt\pi}{2e}\,\frac{\kappa}{C}\,\frac{\sigma_{\rm eff}}{d} \quad\text{at}\quad t_*=\frac{1}{2\Gamma_{\rm sat}}. \label{eq:law} \end{equation}\] For Gaussian sheets at \(C=1\) this is \(8.8\times10^{-3}\,\sigma_{\rm eff}/d\), and the slab ceiling would give \(2.2\times10^{-2}\,\sigma_{\rm eff}/d\) if the law holds. At finite \(\xi\), \(n\) times the peak is \(\xi/2e-0.239+O(1/\xi)\), and it vanishes at \(\xi=1\). We have derived Eq. \(\eqref{eq:law}\); we have not bounded its error. Compared with exact partial transposes of uniform lattice sheets at \(C=1\) it agrees to a few per cent; at \(C=2\) it overestimates the exact peak. Because \(\mathcal{R}\) multiplies a dephasing time, “negativity of order \(10^{-2}\,\sigma/d\)” is the statement that the entangling rate is a fraction of the dephasing rate, not an extra suppression.
An explicit state. Take for each mass a hexagonal patch of 19 point branches with spacing \(8\,\sigma\), plus its far copy (38 branches per mass), the two patches facing each other at slab separation \(d=3.52\,\sigma\), beyond the two-branch ceiling of Proposition 3.2, and \(C=1\). Then \(\rho_{\max}=0.813>s_*\), \(\kappa=0.0258\), \(\mathcal{R}=0.0130\) and \(\mathcal{R}/2e=2.4\times10^{-3}\). Exact diagonalisation of the \(1444\)-dimensional partial transpose gives a smallest eigenvalue of \(-1.6\times10^{-3}\) at \(t'=0.5\), when each body’s coherence is still \(0.75\); the peak lies below the large-\(\xi\) value because this state is sparse. Unitary Newtonian gravity (\(1/r\) coupling, no noise) on the same state reaches \(-0.41\).
For extended bodies Eq. \(\eqref{eq:law}\) holds with \(\sigma_{\rm eff}\); for spheres with \(\sigma\ll R\) the negativity of sheet states is floored by the radius, \(\sigma_{\rm eff}=0.470\,R\), however small \(\sigma\) is.
In the design of [1] the branches of \(A\) sit at \(0\) and \(250~\mu\)m, those of \(B\) at \(450\) and \(700~\mu\)m, collinear, with closest approach \(d=200~\mu\)m and \(m=10^{-14}\) kg (a diamond sphere of radius \(R=0.879~\mu\)m). The configuration is parallel (Theorem 3.1), and Eq. \(\eqref{eq:twobranch}\) shows that the model entangles it if and only if \(\sigma>524.494~\mu\)m at \(C=1\) (\(617\), \(910\) and \(2158~\mu\)m at \(C=1.25\), \(2\), \(5\)). At \(\sigma=524~\mu\)m the smeared force between two point masses \(1\) mm apart is \(0.39\) of its Newtonian value; torsion-balance tests of the inverse-square law at such separations show no deviation at that level [17]. No two-branch geometry with closest approach \(200~\mu\)m is entangled unless \(\sigma>73.0~\mu\)m (straddle) or \(90.5~\mu\)m (slab), according to the numerical suprema of Section 3, and Proposition 3.2 guarantees \(\sigma>52.9~\mu\)m (any geometry) or \(72.7~\mu\)m (slab).
Many-branch states at the same separation are entangled for any \(\sigma\), but at the rates of Section 4. For \(\sigma\ll R\) and \(C=1\) each body dephases at \(\Gamma_{\rm sat}=86~\mathrm{s}^{-1}\) (coherence time \(11.6\) ms; over the \(2.5\) s of the proposed experiment the dephasing exponent is \(216\)). A Gaussian sheet state of width \(b\approx630~\mu\)m at \(d=200~\mu\)m entangles at \(\Gamma_{\rm ent}=\kappa_GGm^2/(\hbar d)=8.6\times10^{-3}~\mathrm{s}^{-1}\), so \(\mathcal{R}=9.9\times10^{-5}\), and the law \(\eqref{eq:law}\) puts the peak negativity at \(1.8\times10^{-5}\), \(5.8\) ms after preparation. Unitary Newtonian gravity gives the two-branch design an entangling phase of \(0.314\) rad after \(2.5\) s.
Remains true. For two-branch states with ideal branches the model produces no entanglement at any time unless \(\sigma\) is comparable to the separation: below \(2.2109\,\sigma\) for parallel superpositions (proven), below \(2.75\,\sigma\) at slab separation for any geometry (computer-assisted). In any geometry with branches more than a few \(\sigma\) apart the negativity stays below \(0.191\,\sigma_{\rm eff}/d\) at \(C=1\) (Theorem 4.2), against about \(0.4\) for unitary gravity. An entanglement signal of the size the quantum prediction gives, or one that oscillates or persists beyond a few dephasing times \(1/\Gamma_{\rm sat}\), is incompatible with the model. Before entanglement is at issue, the survival of each body’s superposition over the experiment already tests the model.
No longer true. “The Diósi–Penrose dynamics cannot entangle masses further apart than \(0.85\,\sigma\) (or \(2.21\,\sigma\))”, read as a statement about the model rather than about a two-branch geometry; equivalently, “the dynamics is separability-preserving for \(d\gg\sigma\)”. And the geometry-free inference “any entanglement detected beyond \(2.2\,\sigma\) falsifies the model, whatever the state and whatever its size”. The inference has to go through the size and the time course of the signal, or be restricted to two-branch geometries. For the design above, with ideal branches, a positive result of the expected size excludes the model for every \(\sigma\) below \(524~\mu\)m. For wavepackets of finite width the criterion is degenerate (Remark 2.3) and we give no bound on what they produce, so the statement is not extended to any positive result.
Answer to the question of [9]: entanglement from this model can in principle be produced at any separation, with superpositions over many branches whose coherent extent is several times the separation; it is then produced at a rate of order \(\sigma/d\) times the rate at which the superposition itself is destroyed, and, for branches more than a few \(\sigma\) apart, it never exceeds a fraction of order \(\sigma/d\) of what unitary gravity gives.
Position-eigenstate branches. Theorem 2.2 is exact for branches that are position eigenstates. For square-integrable wavefunctions \(\rho_{\max}=1\) (Remark 2.3): every pair of real wavepackets is entangled at once, and the yes/no question has no content there. For wavefunctions Theorem 7.1 gives the entangling direction, with the rate \(\eqref{eq:rate}\) in integral form; the negativity bound \(\eqref{eq:negbound}\) is proven for finite branch sets only, and we give no bound in terms of the packet width.
The kinetic term. All results are for \(H=0\). The initial rate \(\eqref{eq:rate}\) and the bounds of Theorem 4.1 are unchanged by any local Hamiltonian defined on the initial state, and local Hamiltonians acting within a finite set of sites cannot make a sub-threshold region entangling (Theorem 2.2(b)). Beyond that we control nothing: free packets acquire full support at once and the negativity bound has not been extended to them. For \(m=10^{-14}\) kg the relevant experiments sit deep in the static regime, which is an argument, not a bound.
Preparation. Every result assumes the product state exists at \(t=0\). The same dephasing acts while it is made: a superposition over at least about \(15\,d/\sigma\) positions, more than five gaps across, must be prepared in a time short compared with \(1/\Gamma_{\rm sat}\) (\(11.6\) ms for the mass above), or it never exists.
What is numerical. The critical distances, the constants \(\Lambda_+\), \(L_{1/2}\), the extent thresholds and the rate constants are roots and maxima of explicit one-dimensional functions, or top eigenvalues of converged discretisations of Love’s operator. Proposition 3.2 is computer-assisted, in floating point, without interval arithmetic, with a margin of \(0.36\%\). That the two-branch suprema over all geometries equal the closed forms of Table 1 rests on numerical search. The optimality of the axisymmetric sector on discs, and the identification of the best single-disc pattern as dipolar, are numerical. The leading-order law \(\eqref{eq:law}\) is derived without an error bound and was compared with exact partial transposes of lattice sheets only; the explicit state of Section 4.4 is one exact instance. Exact partial transposes at larger separations and with more branches agree with the law at \(C=1\); we do not quote them.
Also open. Separability after the negative eigenvalue has returned to zero; the best rate constant, bracketed between \(0.027\) and \(0.068\) at \(C=1\); non-planar supports; noise kernels other than the Newtonian one; \(C<1\), which is not a classical model; the variant with an unsmeared force, which is not a measurement-and-feedback model and to which the pair decomposition does not apply.
Discussion. The Diósi–Penrose model is a classical channel with a definite structure: one kernel in force and noise, and a local dephasing equal to the Gram form of that kernel. That structure gives an exact criterion, a canonical correlation against \(C/\sqrt{1+C^2}\), and it ties the entangling rate to the model’s own dephasing rate by a factor of order \(\sigma/d\). The model’s generator is not separability-preserving at any distance, so it is not a local-operations-and-classical-communication channel. But what it produces at separations large compared with \(\sigma\) is small and short-lived, and it is preceded by the loss of each body’s coherence at the Diósi–Penrose rate. For entanglement-witness experiments the useful discriminants are the size and the time course of the signal, and, before them, the survival of the single-body superposition.
Absorb the masses into \(g\). Choose orthonormal bases \(e_i\) of \(V_A\) and \(e_i'\) of \(V_B\) with \(\langle e_i,e_j'\rangle=s_i\delta_{ij}\) (singular value decomposition; \(s_1=\rho_{\max}\), \(s_i=0\) beyond the smaller rank). Fix reference branches \(a_0\), \(b_0\) and put \(x_i(a)=\langle e_i,g_a-g_{a_0}\rangle\), \(y_i(b)=\langle e_i',g_b-g_{b_0}\rangle\). Since the \(g_a-g_{a'}\) lie in \(V_A\), \[\begin{equation} D_A=\tfrac12\sum_i\big(x_i(a)-x_i(a')\big)^2,\qquad \delta=\sum_is_i\big(x_i(a)-x_i(a')\big)\big(y_i(b)-y_i(b')\big), \end{equation}\] and \(F(a,b)=\sum_is_ix_i(a)y_i(b)+\varphi_A(a)+\varphi_B(b)+\text{const}\). Hence \(\Lambda=\sum_i\Lambda_i+i[\varphi_A(a)-\varphi_A(a')]+i[\varphi_B(b)-\varphi_B(b')]\) with \(\Lambda_i\) as in Eq. \(\eqref{eq:pairs}\); the phases \(\varphi\) are local unitaries. For each \(i\) take real Gaussian variables \((\eta_i,\zeta_i,\eta_i',\zeta_i')\), mean zero, independent for different \(i\), with covariance \[\begin{equation} t\,s_i\begin{pmatrix}|J|&J\\J&|J|\end{pmatrix},\qquad J=\begin{pmatrix}0&\tfrac12\\ \tfrac12&C\end{pmatrix}, \end{equation}\] positive semidefinite because \(|J|\pm J\geq0\). Define the random diagonal operators \(U_A=\sum_a\exp[\sum_ix_i(a)(\eta_i+i\zeta_i)]\,|a\rangle\langle a|\) and \(U_B\) likewise with \(y_i\), \(\eta_i'\), \(\zeta_i'\). With \(\mathbb E\,e^Z=e^{\mathbb E[Z^2]/2}\) the cross covariances \(\mathbb E[\zeta_i\zeta_i']=ts_iC\) and \(\mathbb E[\eta_i\zeta_i']=\mathbb E[\zeta_i\eta_i']=ts_i/2\) give \[\begin{equation} \mathbb E\big[u_A(a)u_B(b)\bar u_A(a')\bar u_B(b')\big] =\exp\Big\{t\sum_is_i\big[i(x_iy_i-x_i'y_i')-C(x_i-x_i')(y_i-y_i')\big]\Big\}\,L_A(a,a')L_B(b,b'), \end{equation}\] using \((x+x')(y-y')+(x-x')(y+y')=2(xy-x'y')\), with local factors \(L\). Dividing, \(e^{t\Lambda}=\mathbb E[u\bar u]\,T_A(a,a')T_B(b,b')\) up to the local phases, with \[\begin{equation} T_A(a,a')=d(a)\overline{d(a')}\,\exp\Big\{t\sum_i\big(C-s_i\operatorname{tr}|J|\big)x_i(a)x_i(a')\Big\}, \qquad\operatorname{tr}|J|=\sqrt{1+C^2}, \end{equation}\] \(d\) a non-vanishing diagonal factor, and the same for \(T_B\). If \(s_i\sqrt{1+C^2}\leq C\) for all \(i\) the exponent is \(t\) times a positive semidefinite matrix; its entrywise exponential is positive semidefinite by the Schur product theorem [15], so \(T_A\) and \(T_B\) act as completely positive Schur multipliers \(\mathcal T_A\), \(\mathcal T_B\). Therefore \[\begin{equation} \rho_t=\mathbb E\Big[(U_A\otimes U_B)\,(\mathcal T_A\otimes\mathcal T_B)(\rho_0)\,(U_A\otimes U_B)^\dagger\Big], \end{equation}\] a mixture of products of local completely positive maps, trace preserving in total because the diagonal of \(e^{t\Lambda}\) is one. Separable states go to separable states. Interleaved local channels and the Lie–Trotter limit in finite dimension, with the separable set closed, give the last statement. The construction needs only that the Gram matrices \(f_A\), \(f_B\) are non-singular (distinct branches); it holds for any number of branches, any amplitudes and every \(t\geq0\). If \(\rho_{\max}>s_*\), then \(T_A\) and \(T_B\) are indefinite and part (a) shows the state entangled. 0◻
Theorem 7.1. Let \(\psi_A,\psi_B\in L^2(\mathbb R^3)\) and \(H=0\). Suppose there are bounded neutral \(q_1\), \(q_2\), supported on sets of finite measure on which \(|\psi_A|,|\psi_B|\geq\varepsilon>0\), with \(C\langle q_1,q_1\rangle+C\langle q_2,q_2\rangle-2\,\mathrm{Re}[(C+i)\langle q_1,q_2\rangle]<0\) in the smeared Coulomb form. Then \(\rho_t\) is entangled for all sufficiently small \(t>0\).
Proof. The kernel \(\Lambda\) is bounded (\(|f|\leq f_0\)). With \(c=q_1\otimes|\psi_B|^2+|\psi_A|^2\otimes q_2\in L^1\) and \(\chi=c/\tilde\Psi^*\in L^2\), \(\langle\chi|\rho_t^{T_1}|\chi\rangle=\int\!\!\int\bar c\,c\,e^{t\Lambda} =t\,(\text{negative})+O(t^2\|c\|_1^2)\). A separable state has a positive partial transpose, whose quadratic form is non-negative. ◻
Since the smeared point charges of any open set span the Coulomb space (their potentials are real-analytic), such \(q_1\), \(q_2\) exist for every pair of supports with interior and every separation.
Lengths are in units of \(\sigma\) unless stated.
\(h(x)=f(\sqrt x)=\pi^{-1/2}\int_0^1e^{-xu^2/4}du\) is a mixture of decaying exponentials: decreasing, convex and log-convex. Put \(x=|r|^2\), \(y=|w|^2\), \(P=|r-w|^2\), \(Q=|r+w|^2\), so \(P+Q=2x+2y\), and \(N=2h(x)-h(P)-h(Q)\), \(s=|N|/(2D(|w|))\). If \(N>0\): convexity gives \(h(P)+h(Q)\geq2h(x+y)\), log-convexity \(h(x+y)h(0)\geq h(x)h(y)\), so \(N\leq2[h(x)-h(x+y)]\leq2(f(|r|)/f_0)\,D(|w|)\), and \(|r|\geq d\). If \(N<0\): (i) \(-N\) is the second difference of \(t\mapsto f(|r+tw|)\) at \(t=-1,0,1\), hence \(\leq|w|^2\Lambda_+\), with \(\Lambda_+\) the largest Hessian eigenvalue of \(f\) over space, which is \(\max f''\) because \(f'<0\); (ii) with \(P\leq Q\) the parallelogram law gives \(Q\geq x\), so \(-N\leq h(P)-h(x)\leq f(d)\). For \(|w|\geq L_{1/2}\), (ii) gives \(-N/(2D)\leq f(d)/f_0\). For \(|w|\leq L_{1/2}\), (i) gives \(-N/(2D)\leq\Lambda_+|w|^2/(2D(|w|))\leq\Lambda_+L_{1/2}^2/f_0\), because \(L^2/D(L)\) is increasing (\(D(L)/L^2\) is a difference quotient of the convex \(h\)). The transversal family with \(L\to\infty\) approaches \(f(d)/f_0\) from below. 0◻
Write \(N=\psi(b_1)-\psi(b_2)\) with \(\psi(b)=f(|b-a_1|)-f(|b-a_2|)\). (i) \(|\psi|\leq g_dL_A\) with \(g_d=\sup_{r\geq d}|f'|\). (ii) \(|\psi|\leq m_d(L_A)\), \(m_d(L)=\sup_{r\geq d}[f(r)-f(r+L)]\). The same with \(A\) and \(B\) exchanged. (iii) \(N\) is a double difference, so \(|N|\leq L_AL_B\sup\|\nabla^2f\|\) over the parallelogram of difference vectors, on which \(|x|\geq d\) at slab separation and \(|x|\geq d-(L_A+L_B)/2\) at closest approach; \(\|\nabla^2f\|= \max(|f''|,|f'|/r)\leq\kappa_0\). Then \[\begin{equation} s\leq\frac{\min\{2g_dL_A,\,2g_dL_B,\,2m_d(L_A),\,2m_d(L_B),\,HL_AL_B\}}{2\sqrt{D(L_A)D(L_B)}}, \end{equation}\] with \(H\) the supremum in (iii), maximised over \((L_A,L_B)\) with cell bounds (numerators at the upper corner, denominators at the lower corner of each cell of a geometric grid of ratio \(1.01\), a last cell for \(L>400\,\sigma\)), using that \(L/\sqrt{D(L)}\), \(m_d\) and \(D\) are monotone. The suprema over \(r\) are taken on grids. The bound is non-increasing in \(d\) and crosses \(1/\sqrt2\) where stated in Section 3. 0◻
Smearing lowers the self-energy, so \(\langle q,q\rangle\leq16/(3\pi a)\) per unit-charge uniform disc; \(f(r)\geq\operatorname{erf}(d/2\sigma)/r\) for \(r\geq d\); and for unsmeared coaxial discs \(E(0)-E(d)=\int\frac{d^2k}{(2\pi)^2}\frac{2\pi}{k}|\hat q|^2(1-e^{-kd})\leq2\pi d\int q^2 =2d/a^2\). The far copy makes the pattern neutral without changing the ratio; this proves \(\eqref{eq:discbound}\). For the first statement at arbitrary \(d\) and \(C\), let \(A\) carry a uniform planar lattice patch (with in-plane distances \(\Delta_{jj'}\)) and its far copy, and \(B\) the same patch translated by \(d\) along the normal. The same Fourier estimate, with the smeared kernel, gives \(1-\rho_{\max}\leq(d/2\sigma^2)\sum_{jj'}e^{-\Delta^2_{jj'}/4\sigma^2}/\sum_{jj'}f(\Delta_{jj'})\); the numerator grows like the number of points and the denominator like its square times \(f\) of the patch size, so the right side tends to zero for growing patches. As wavefunctions, discs of small thickness inherit a strict inequality by continuity of the bounded smeared kernel. 0◻
For charges on coaxial unit discs at gap \(\delta\) (\(\sigma\to0\)), the mutual energy is, up to a common constant, \(E(\delta)=\int_0^\infty e^{-k\delta}\hat q(k)^2dk\), with \(\hat q\) the Hankel transform of the charge. Writing an axisymmetric charge through its Abel transform, \(\hat q(k)=\int_0^1g(t)\cos(kt)\,dt\) with \(g\) extended evenly to \((-1,1)\), gives \(E(0)=(\pi/2)\int_0^1g^2\) and \(E(\delta)=(\pi/2)\int_0^1g\,T_\delta g\), with \(T_\delta\) of Eq. \(\eqref{eq:love}\), the operator of Love’s integral equation [16]. The canonical correlation is the top eigenvalue of \(T_\delta\) on even functions. The uniform disc is \(g(t)=\sqrt{1-t^2}\), for which the Rayleigh quotient is \(\Phi\) of Eq. \(\eqref{eq:Phi}\). 0◻
(i) \(C\langle q_1,q_1\rangle+C\langle q_2,q_2\rangle\geq2C|\langle q_1,q_2\rangle|/\rho_{\max}\), so the numerator of \(\eqref{eq:rate}\) is \(\geq-2(1-\theta)\sqrt{1+C^2}|\langle q_1,q_2\rangle|\), and \(|\langle q_1,q_2\rangle|\leq f(d)\|q_1\|_1\|q_2\|_1\leq\tfrac12f(d)(\sum|q_1|^2/p+\sum|q_2|^2/r)\) by Cauchy–Schwarz. (ii) For charges at depths \(z\leq0\) and \(z'\geq d\), \(\langle q_1,q_2\rangle=\int\frac{d^2k}{(2\pi)^2}\frac{2\pi}{k} e^{-kd}\bar A_1A_2\) with \(A_1(k)=\int dz\,e^{kz}\hat q_1(k,z)\), \(|A_1|\leq\|q_1\|_1\), and \(\langle q_1,q_1\rangle\geq\int\frac{d^2k}{(2\pi)^2}\frac{2\pi}{k}|A_1|^2\), because \(e^{-k|z-z'|}-e^{k(z+z')}\) is \(2k\) times the Dirichlet Green function of \(-\partial_z^2+k^2\) on the half-line, a positive kernel. Then mode by mode the numerator is at least \(-\int\frac{d^2k}{(2\pi)^2}\frac{2\pi}{k}[\sqrt{1+C^2}e^{-kd}-C]_+(|A_1|^2+|A_2|^2)\), and \(\int_0^\infty[\sqrt{1+C^2}e^{-kd}-C]_+dk=B(C)/d\). With smearing, for planar supports, \(e^{-kz}\) is replaced by \(w(k,z)=\tfrac12[e^{-kz}\operatorname{erfc}(\sigma k-z/2\sigma)+e^{kz}\operatorname{erfc}(\sigma k+z/2\sigma)]\), which gives the finite-\(\sigma\) value quoted. Eq. \(\eqref{eq:kappaG}\) is the rate of the witness proportional to the sheet profile, evaluated with the same mode formula. 0◻
Let \(\Lambda_\theta\) be \(\Lambda\) with \(F\) replaced by \(\theta F\) in the Hamiltonian and in the correlated noise, local dephasing unchanged, and \(\rho_t^\theta=\rho_0\circ e^{t'\Lambda_\theta}\). Its canonical correlations are \(\theta s_i\leq s_*\), so by Remark 2.1 \(\rho_t^\theta\) is separable and \((\rho_t^\theta)^{T_1}\geq0\). Now \(\rho_t=\rho_t^\theta\circ M\) with \(M=\exp[t'(1-\theta)(i\epsilon-C\delta)]\), so by Weyl’s inequality \(\lambda_{\min}(\rho_t^{T_1})\geq-\|\rho_t^\theta\circ(M-1)\|_{\rm HS}\), the Hilbert–Schmidt norm being invariant under partial transposition. Entrywise \(|\rho_t^\theta|=(p_ap_{a'}r_br_{b'})^{1/2} e^{-t'C(D_A+D_B+\theta\delta)}\) and \(|M-1|\leq t'(1-\theta)(\epsilon^2+C^2\delta^2)^{1/2} \max(1,e^{-t'(1-\theta)C\delta})\); every exponent is non-positive since \(D_A+D_B+\delta=\tfrac12\|g_a-g_{a'}+g_b-g_{b'}\|^2\geq0\). For the corollary: terms with \(a\neq a'\), \(b\neq b'\) have \(C^2\delta^2+\epsilon^2\leq4C^2f(d)^2\) (for \(C\geq1/\sqrt3\)) and exponent \(\geq2D_\ell-2f(d)\); terms with one index pair equal have \(\delta=0\), \(|\epsilon|\leq f(d)\), exponent \(\geq D_\ell\) and weight \(\sum p^2\) or \(\sum r^2\); and \(\sup_tte^{-\alpha t}=1/(e\alpha)\). For \(\ell,d\gg\sigma\), \(D_\ell\to f_0=1/\sqrt\pi\) and \(\sum p^2,\sum r^2\to0\). 0◻
[1] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn, Spin Entanglement Witness for Quantum Gravity, Phys. Rev. Lett. 119, 240401 (2017). doi:10.1103/PhysRevLett.119.240401
[2] C. Marletto and V. Vedral, Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity, Phys. Rev. Lett. 119, 240402 (2017). doi:10.1103/PhysRevLett.119.240402
[3] L. Diósi, A universal master equation for the gravitational violation of quantum mechanics, Phys. Lett. A 120, 377 (1987). doi:10.1016/0375-9601(87)90681-5
[4] L. Diósi, Models for universal reduction of macroscopic quantum fluctuations, Phys. Rev. A 40, 1165 (1989). doi:10.1103/PhysRevA.40.1165
[5] R. Penrose, On gravity’s role in quantum state reduction, Gen. Relat. Gravit. 28, 581 (1996). doi:10.1007/BF02105068
[6] A. Tilloy and L. Diósi, Sourcing semiclassical gravity from spontaneously localized quantum matter, Phys. Rev. D 93, 024026 (2016). doi:10.1103/PhysRevD.93.024026
[7] A. Tilloy and L. Diósi, Principle of least decoherence for Newtonian semiclassical gravity, Phys. Rev. D 96, 104045 (2017). doi:10.1103/PhysRevD.96.104045
[8] D. Kafri, J. M. Taylor, and G. J. Milburn, A classical channel model for gravitational decoherence, New J. Phys. 16, 065020 (2014). doi:10.1088/1367-2630/16/6/065020
[10] O. Angeli and M. Carlesso, Entanglement in Markovian hybrid classical-quantum theories of gravity, Phys. Rev. D 112, 024047 (2025). doi:10.1103/jzht-fbwt
[11] A. Di Biagio, Gravity-induced entanglement is not a theory-independent witness of nonclassicality of the gravitational field, arXiv:2511.02683 (2025). arXiv:2511.02683
[12] Y.-X. Wang, J. S. Pedernales, and M. B. Plenio, Recoverability of Gravitational Entanglement under Decoherence, arXiv:2611.05044 (2026). arXiv:2611.05044
[13] H. Hotelling, Relations Between Two Sets of Variates, Biometrika 28, 321 (1936). doi:10.1093/biomet/28.3-4.321
[14] A. Peres, Separability Criterion for Density Matrices, Phys. Rev. Lett. 77, 1413 (1996). doi:10.1103/PhysRevLett.77.1413
[15] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge University Press, 2013). ISBN 9780521839402.
[16] E. R. Love, The Electrostatic Field of Two Equal Circular Co-axial Conducting Disks, Q. J. Mech. Appl. Math. 2, 428 (1949). doi:10.1093/qjmam/2.4.428
[17] J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer, and B. R. Heckel, New Test of the Gravitational \(1/r^2\) Law at Separations down to 52 \(\mu\)m, Phys. Rev. Lett. 124, 101101 (2020). doi:10.1103/PhysRevLett.124.101101