\section{The model and the criterion}
\label{sec:criterion}

\subsection{The master equation in the position basis}

The Tilloy--Di\'{o}si dynamics with smeared force and noise, in the notation
of~\cite{TrilloNavascues2025} 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; both
carry the same density and the same kernel. In measurement-and-feedback realisations $C\geq1$, and $C=1$ is
the least-decoherence point~\cite{TilloyDiosi2016,TilloyDiosi2017}, the case
of~\cite{TrilloNavascues2025}.

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 $\ip{u}{v}=\int\!\!\int u(x)v(y)/|x-y|$ and put
\begin{equation}
  f_A(a,a')=\ip{g_a}{g_{a'}},\qquad f_B(b,b')=\ip{g_b}{g_{b'}},\qquad F(a,b)=\ip{g_a}{g_b}.
  \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, $\|u\|^2=\ip{u}{u}$. Expanded, the real part is
$-C\,[m_A^2D_A+m_B^2D_B+m_Am_B\,\delta]$ with $D_A=f_A(a,a)-f_A(a,a')$, $D_B$ likewise, and
$\delta=F(a,b)+F(a',b')-F(a,b')-F(a',b)$. The local terms $D_A$, $D_B$ are the Di\'{o}si--Penrose
dephasing of each body; $\delta$ is the correlated part of the noise.

\subsection{The criterion}

A state is a \emph{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~\ref{sec:discussion} discusses wavepackets). We use two
measures of separation: the \emph{closest approach}, the smallest distance between a branch of $A$ and a
branch of $B$; and the \emph{slab separation} $d$, for which some plane has all branches of $A$ on one side
and all branches of $B$ on the other, at normal distance at least $d$. The second implies the first and
forbids one body from sitting between the branches of the other.

\begin{definition}
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}
  \rmax(S_A,S_B)=\sup\Big\{\frac{|\ip{u}{v}|}{\|u\|\,\|v\|}:\ u\in V_A,\ v\in V_B\Big\},
  \qquad \sth(C)=\frac{C}{\sqrt{1+C^2}} .
  \label{eq:rhomax}
\end{equation}
\end{definition}

$\rmax$ is the cosine of the smallest angle between $V_A$ and $V_B$: 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.
$\sth(1)=1/\sqrt2=0.7071$. The theorem says that these two numbers decide the question, in both directions
and for all times.

\begin{theorem}[Entanglement criterion]
\label{thm:criterion}
Let $H=0$ and $C>0$, and let $S_A$, $S_B$ be finite sets of distinct branches.
\begin{enumerate}
\item[(a)] If $\rmax>\sth$, 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\ip{q_1}{q_1}+C m_B^2\ip{q_2}{q_2}-2m_Am_B\,\mathrm{Re}\big[(C+i)\ip{q_1}{q_2}\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
$\ip{q_1}{q_2}=\sum_{ab}\bar q_1(a)q_2(b)F(a,b)$ and $\ip{q_1}{q_1}=\sum\bar q_1(a)q_1(a')f_A(a,a')$.
\item[(b)] If $\rmax\leq\sth$, 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.
\item[(c)] 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 $\rmax>\sth$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
(a) Partial transposition, written $T_1$, 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 $|\ip{q_1}{q_2}|^2>\sth^2\ip{q_1}{q_1}\ip{q_2}{q_2}$, that is, if and
only if $\rmax>\sth$; the state is then entangled by the Peres criterion~\cite{Peres1996}.

(b) Choose bases of $V_A$ and $V_B$ that diagonalise the cross form (a singular value decomposition with
singular values $s_i\leq\rmax$) 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}
  \begin{split}
  \Lambda&=\sum_i\Lambda_i+(\text{local phases}),\\
  \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'),
  \end{split}
  \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~\ref{app:proofs} gives the covariance and the bookkeeping. (c) follows from (a) and (b).
\end{proof}

The criterion depends on the branch sets, $\sigma$, the shapes of the bodies and $C$, not on the masses,
the amplitudes or the time. A pair of patterns $(q_1,q_2)$ that makes the numerator of
Eq.~\eqref{eq:rate} negative is called a \emph{witness}. The threshold is below one because
$|C+i|=\sqrt{1+C^2}>C$. 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.

\begin{remark}[Generality]
\label{rem:general}
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~\ref{thm:criterion} therefore holds for any real cross kernel
$\bar F$ with $\rmax$ 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~\ref{thm:negativity}
below uses this for $\bar F=\theta F$.
\end{remark}

\begin{remark}[Relation to Wang, Pedernales and Plenio]
\label{rem:wpp}
Each $\Lambda_i$ in Eq.~\eqref{eq:pairs} is a generator of the form treated in Lemma~3
of~\cite{WangPedernalesPlenio2026}, and their condition for $e^{t\Lambda_i}$ to be separable is
$s_i\sqrt{1+C^2}\leq C$ (Appendix~\ref{app:proofs} gives the dictionary). 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. 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 is part (a).
\end{remark}

\begin{remark}[The idealisation]
\label{rem:ideal}
Potentials of smeared charges are real-analytic, so the smeared point charges of any open set span the
whole Coulomb space: $\rmax=1$ for any two supports with interior. Every pair of wavefunctions whose
supports have interior is therefore entangled at once, at every separation
(Theorem~\ref{thm:continuum}). The ``if and only if'' of
Theorem~\ref{thm:criterion} is a statement about position-eigenstate branches, the idealisation also used
in~\cite{TrilloNavascues2025,WangPedernalesPlenio2026}. The physical content lies in the rate and the size
of the entanglement, Section~\ref{sec:rate}.
\end{remark}