\section{Noise robustness}\label{sec:ppt3-noise}

How much white noise or independent qubit depolarisation can the states tolerate while remaining GME?
We bound the tolerance from below with exact symmetric-extension witnesses, and from above with explicit biseparable
decompositions completed by the Gurvits--Barnum separable ball~\cite{GB}. The bounds meet to within a few per cent in
relative terms. Appendix~\ref{sec:ppt3-app-noise} gives the certificate checks.

The supported witnesses establish GME on a five-dimensional support.
Mixing in a small amount of white noise yields full-rank, strictly PPT
states while preserving GME, giving the following general consequence.

\begin{corollary}[positive volume in any finite dimension]\label{cor:ppt3-volume}
\cls{proved; analytic consequence of Theorem~\ref{thm:ppt3-family}}
In any finite multipartite system, if a fully-PPT GME state
exists, such states have non-empty interior (and hence positive
volume) among density matrices. In particular this holds for
three qubits. The conclusion is analytic; the certificates below
provide explicit noise and preparation radii.
\end{corollary}
\begin{proof}
Let $\rho$ be any fully-PPT GME state on a Hilbert space of total
dimension $d$. The biseparable set $\mathrm{BS}$ is compact, being the
convex hull of the union of finitely many compact sets of pure
cut-product states. Since $\rho\notin\mathrm{BS}$, the noisy state
$\rho_p=(1-p)\rho+pI/d$ remains outside $\mathrm{BS}$ for all
sufficiently small $p>0$. But $\rho_p\succeq pI/d$ and
$\rho_p^{T_S}=(1-p)\rho^{T_S}+pI/d\succeq pI/d$ at every cut $S$.
Thus $\rho_p$ is full rank and strictly PPT, and it is GME.
Continuity gives an open neighbourhood of $\rho_p$ in the trace-one
Hermitian affine space consisting entirely of physical fully-PPT GME
states. Applying this observation to Theorem~\ref{thm:ppt3-family}
proves the three-qubit claim.
\end{proof}

Positive volume does not say how large the margins are. Since PPT tests cannot detect these states, certifying an
explicit margin requires a stronger relaxation of biseparability. We use dual witnesses
from level two of the Doherty--Parrilo--Spedalieri (DPS) symmetric-extension hierarchy~\cite{DPS02,DPS,DPS05}, the lowest level
stronger than PPT. Applying it across each cut to relax biseparability was proposed in Appendix~B of~\cite{JMG}.
Every cut-product vector extends by duplicating its qubit factor; the following certificate exploits this necessary
condition. Numerical searches supplied candidates, rounded to rational parameters; only exact checks establish the
bounds (Appendix~\ref{sec:ppt3-verifiers}).
Here $\mathrm{Sym}^2(A)$ is the subspace invariant under exchanging the two copies of $A$; tensor subscripts specify
which factors an operator acts on.

\begin{lemma}[certificate and stability principle]\label{lem:ppt3-dps}
\cls{proved}
Let $A|B$ be a bipartition into finite-dimensional factors, let $V$ have full column rank with image
$\mathrm{Sym}^2(A)\otimes B$ inside $A_1\otimes A_2\otimes B$ (the columns need not be normalised), and let $Y$ be
Hermitian on $A\otimes B$ and $Z_1,Z_2\succeq0$ on $A_1\otimes A_2\otimes B$ with
\begin{equation}\label{eq:ppt3-cert}
P=V\dg\bigl(Y_{A_1B}\otimes I_{A_2}-Z_1^{T_B}-Z_2^{T_{A_2}}\bigr)V\succeq0 .
\end{equation}
Then $\langle x|Y|x\rangle\ge0$ for every product vector $x=a\otimes b$. Consequently, on three qubits, if $Y$ is invariant
under cyclic permutations of the qubits and satisfies \eqref{eq:ppt3-cert} for the cut $A|BC$, then
$\tr(Y\sigma)\ge0$ for every biseparable state $\sigma$. If moreover $\rho$ is a three-qubit state with
$\tr(Y\rho)=-g<0$, $h=\tr(Y)/8>0$, and $\operatorname{spec}(Y)\subset[l,u]$, then
\begin{equation}\label{eq:ppt3-stability}
(1-p)\rho+pI/8\ \text{is GME for }p<\frac g{g+h},\qquad D(\rho,\mathrm{BS})\ge\frac g{u-l}.
\end{equation}
Finally, if $\rho^{T_j}\succeq cI$ at every cut $j$, with positive $c$, then every
physical state $\sigma$ with
\[
D(\rho,\sigma)<\frac{2c}{1+\sqrt2}
\]
is strictly PPT across every cut. This constant is dimension independent.
\end{lemma}

\emph{Idea of the proof.} For a product vector $x=a\otimes b$, the certificate writes $\|a\|^2\langle x|Y|x\rangle$ as the sum of a positive
semidefinite quadratic form (from $P$) and two nonnegative terms (from $Z_1$ and $Z_2$), so $Y$ is nonnegative on every cut-product vector and hence on $\mathrm{BS}$.
The noise and trace-distance bounds follow from the sign and spectral width of $Y$. The PPT-stability radius uses that the partial transpose of
a pure state has spectral width at most $(1+\sqrt2)/2$. Details in Appendix~\ref{sec:ppt3-app-noise}.

Local depolarisation means applying $\mathcal D_{q_j}(X)=(1-q_j)X+q_j\tr(X)I/2$ on qubit $j$, with
\emph{independent rates} $0\le q_j\le q$. White noise means $(1-p)\rho+pI/8$.

\begin{corollary}[noise and preparation radius of the family]\label{cor:ppt3-noise}
\cls{exact-rational certificate, with proved PPT stability; DS-1, DS-2, DS-3, DS-6}
The rational, symmetry-reduced cyclic witnesses $Y_t$ of the ancillary certificate data give the following conservative tolerances, where the balls are trace-distance balls
around $\rho_t$.
\begin{center}
\small\setlength{\tabcolsep}{5pt}
\begin{tabular}{@{}lcccc@{}}\toprule
$t$ & white noise $p$ & local $q$ per qubit &
 GME ball & fully-PPT ball\\\midrule
$327/100$ & $3.3\times10^{-4}$ & $1.4\times10^{-4}$ &
 $1.0\times10^{-4}$ & $1.01\times10^{-4}$\\
$10/3$ & $8.7\times10^{-4}$ & $3.7\times10^{-4}$ &
 $2.7\times10^{-4}$ & $5.7\times10^{-4}$\\
$169/50$ & $1.4\times10^{-3}$ & $6.1\times10^{-4}$ &
 $4.4\times10^{-4}$ & $2.0\times10^{-4}$\\\bottomrule
\end{tabular}
\end{center}
Specifically, white noise
$(1-p)\rho_t+pI/8$ remains GME for $p$ up to the listed
value, and independent local depolarisation with rates
$q_j\le q$ does likewise. Every physical state in a listed closed GME ball is GME, and every
state strictly inside a listed PPT ball is fully PPT. Every state at trace distance
strictly below the smaller of the last two entries therefore has \emph{both} properties:
below $2.7\times10^{-4}$ for $t=10/3$, below $1.0\times10^{-4}$ for $t=327/100$ (where the
whole GME ball is also PPT) and below $2.0\times10^{-4}$ for $t=169/50$.
In particular $D(\rho_{10/3},\mathrm{BS})>2.7\times10^{-4}$.
White noise itself preserves PPT automatically.

One fixed dual witness also certifies \emph{uniformly}
over the exact interval \eqref{eq:ppt3-interval}:
$p\le2.4\times10^{-4}$ of white noise and a GME ball
of radius $7.5\times10^{-5}$. This gives a second GME proof, independent of the analytic witness.
\end{corollary}

\emph{Idea of the proof.} Exact witnesses satisfy Lemma~\ref{lem:ppt3-dps}. Unequal depolarisation rates reduce to one
uniform rate because further local noise preserves biseparability. Affine dependence on $t$ reduces the uniform
interval bound to two endpoint checks (Appendix~\ref{sec:ppt3-app-noise}).

\begin{corollary}[noise bounds for the surface state from one exact witness]
\label{cor:ppt3-noise-reused}
\cls{exact-symbolic certificate; DS-6}
Let $\rho_*$ be the trace-one normalisation of the analytic
surface operator $\rho(312,266,603)$ in
\eqref{eq:ppt3-rho-surface}; here
$\tau=133/156$ and $a=201/104$.
It is fully PPT and GME, with rank profile $(5,5,5,5)$.
A single exact level-two witness $Y_*$ (optimised numerically for $\rho_*$ under a spectral bound)
certifies:
\begin{enumerate}[label=(\roman*)]
\item The states $(1-p)\rho_*+pI/8$ remain fully PPT
and GME for $0\le p\le2153/500000$
(about $0.43\%$ white noise).
\item Independent local depolarisation preserves both
properties whenever each rate is at most
$81/50000$ (about $0.16\%$ per qubit).
\item Let
\[
 \rho_\circ=\left(1-\frac{32}{10000}\right)\rho_*
                  +\frac{32}{10000}\frac I8 .
\]
Every physical state $\sigma$ with
$D(\sigma,\rho_\circ)<331/1000000$
is fully PPT and GME. Thus a full-rank state
has a simultaneous preparation radius of
$3.31\times10^{-4}$.
\item Every physical state $\sigma$ with
$D(\sigma,\rho_*)\le129/100000$
is GME (a GME-only radius of $1.29\times10^{-3}$).
\end{enumerate}
\end{corollary}

\emph{Idea of the proof.} The certified expectation and spectral bounds of $Y_*$ give each margin by
Lemma~\ref{lem:ppt3-dps}; white noise opens a positive PPT gap for (iii) (Appendix~\ref{sec:ppt3-app-noise}).

\begin{corollary}[a full-rank fully-PPT GME state tolerating $2.3\%$ white noise]\label{cor:ppt3-robust}
\cls{exact-rational certificate, with proved PPT stability; DS-4, DS-6}
The three-qubit density matrix $\tilde\rho$ with seven-decimal rational coordinates in the certificate data (entries in $\mathbb Q(\sqrt{-3})$) has
full rank, is invariant under cyclic permutations of the qubits and under global parity $Z^{\otimes3}$, has every partial
transpose strictly above $2.49\times10^{-5}$ times the identity, and is GME: its rational level-two witness is
nonnegative on every cut-product vector and negative on $\tilde\rho$. Certified tolerances: global white noise
$p\le2.33\times10^{-2}$, independent depolarising noise with every rate at most $1.0\times10^{-2}$, and every state
within trace distance $7.3\times10^{-3}$ of $\tilde\rho$ is GME. The ball of radius $2.0\times10^{-5}$ is also certified
fully PPT, by the verified lower bound $2.49\times10^{-5}$ on the partial transposes and Lemma~\ref{lem:ppt3-dps}.
These are certified lower bounds; Proposition~\ref{prop:ppt3-noise-sharp} gives the matching upper bound for white noise.
\end{corollary}

\emph{Idea of the proof.} Lemma~\ref{lem:ppt3-dps} applied to the rational data of $\tilde\rho$, with exact checks of full rank, symmetry and the
partial-transpose eigenvalue bounds. Details in Appendix~\ref{sec:ppt3-app-noise}.

The state $\tilde\rho$ is obtained from the white-noise mixture $(1-\epsilon)\rho_{\rm opt}+\epsilon I/8$, $\epsilon=2.5\times10^{-4}$, of a
numerically optimised fully-PPT state $\rho_{\rm opt}$ (Remark~\ref{rem:ppt3-robust-structure}) by rounding to rational
entries; white noise provides strict PPT before rounding. Its GME ball is more than five times the GME-only radius of
Corollary~\ref{cor:ppt3-noise-reused}(iv). Its exact coordinates are displayed in Appendix~\ref{sec:ppt3-app-noise}.

For arbitrary physical perturbations, the certified GME radius is $7.3\times10^{-3}$ in trace distance, whereas the
simultaneous PPT radius is only $2.0\times10^{-5}$ because the smallest partial-transpose eigenvalue is small.
The much larger white-noise tolerance therefore does not guarantee stability under generic preparation errors.

\begin{proposition}[the white-noise thresholds are essentially sharp]\label{prop:ppt3-noise-sharp}
\cls{exact-rational/algebraic certificate, computer-assisted; DS-7}
Let $p_{\rm GME}(\rho)=\sup\{p\in[0,1]:(1-p)\rho+pI/8\text{ is GME}\}$, with $p_{\rm GME}(\rho)=0$ for biseparable $\rho$. Then
\begin{align*}
 \frac{2153}{500000}&\le p_{\rm GME}(\rho_*)\le \frac{4319}{1000000},\\
 \frac{87}{100000}&\le p_{\rm GME}(\rho_{10/3})\le \frac{89}{100000},\\
 \frac{233}{10000}&\le p_{\rm GME}(\tilde\rho)\le \frac{5839}{250000},
\end{align*}
with $\rho_*$ and $\tilde\rho$ as in Corollaries~\ref{cor:ppt3-noise-reused} and~\ref{cor:ppt3-robust}. The noisy state is
biseparable at, and for every $p$ above, each upper bound.
\end{proposition}

\emph{Idea of the proof.} The lower bounds are the three corollaries above. For the upper bounds, an explicit separable-across-$A|BC$ operator,
assembled from three orbit representatives and symmetrised cyclically, is subtracted from the noisy state; the residual lies in the
Gurvits--Barnum ball around the identity~\cite{GB}, hence is separable, so the noisy state is biseparable. Details in Appendix~\ref{sec:ppt3-app-noise}.

The certified intervals have relative widths of at most $2.3\%$, so the small margins reflect the states, not the
relaxation. Numerically the agreement is closer: for all three states the optimal level-two bound and the numerical
value of the optimised three-orbit decomposition agree to within $10^{-7}$ (numerical evidence only); the decompositions use three orbits of qubit directions each.

To guide the search for robust states, we use the convex white-noise robustness $\operatorname{rob}$ below,
rather than $p_{\rm GME}$ itself.

\begin{proposition}[an extremal reduction for white-noise robustness]\label{prop:ppt3-noise-extremal}
\cls{proved; general convex duality}
In any finite multipartite system of dimension $d$, let $\mathcal P$ be the set of fully-PPT states, $\mathrm{BS}$ the
biseparable states and $\operatorname{rob}(\rho)=\min\{s\ge0:(\rho+sI/d)/(1+s)\in\mathrm{BS}\}$, so that
$p_{\rm GME}(\rho)=\operatorname{rob}(\rho)/(1+\operatorname{rob}(\rho))$. Then
\begin{equation}\label{eq:ppt3-robust-dual}
 \operatorname{rob}(\rho)=\max\Bigl\{0,\ \max_{W\in\mathrm{BS}^*,\ \tr W=d}-\tr(W\rho)\Bigr\},
\end{equation}
where $\mathrm{BS}^*$ is the cone of Hermitian operators nonnegative on $\mathrm{BS}$. Consequently $\operatorname{rob}$ is convex,
its maximum over $\mathcal P$ is attained at an extreme point of $\mathcal P$, and so is that of the increasing function
$p_{\rm GME}=\operatorname{rob}/(1+\operatorname{rob})$, and for a fixed witness the best state is
the solution of a semidefinite program. The same holds with $\mathrm{BS}$ replaced by any closed convex set of states
containing $\mathrm{BS}$, for example the states passing every cyclic level-two test of Lemma~\ref{lem:ppt3-dps}.
\end{proposition}
\begin{proof}
$I/d$ is interior to the cone generated by $\mathrm{BS}$: it is a full-support average of product pure states, whose
projectors span the Hermitian operators, so no nonzero element of $\mathrm{BS}^*$ vanishes on it. Hence the conic program
defining $\operatorname{rob}$ is strictly feasible, and conic duality gives \eqref{eq:ppt3-robust-dual} with no gap; each
$W\in\mathrm{BS}^*$ with $\tr W=d$ gives $s\ge-\tr(W\rho)$. Since $I/d$ is interior, the slice
$\{W\in\mathrm{BS}^*:\tr W=d\}$ is compact, so $\operatorname{rob}$ is a maximum of affine functions over a compact set and hence
continuous and convex. By Bauer's maximum principle it attains its maximum over the compact convex set $\mathcal P$ at an
extreme point. For fixed $W$,
minimising $\tr(W\rho)$ over $\rho\succeq0$, $\tr\rho=1$, $\rho^{T_S}\succeq0$ is a semidefinite program.
\end{proof}

\begin{remark}[where the robust states sit]\label{rem:ppt3-robust-structure}
\cls{numerical evidence; DS-7 generator, DS-4 generator}
The search for $\tilde\rho$ alternated the two steps of Proposition~\ref{prop:ppt3-noise-extremal}:
optimising the witness for a state and the fully-PPT state for a witness. For states invariant under the cyclic permutation and global parity, the
state is block diagonal over six sectors labelled by parity and cyclic charge (the partial-transpose constraints mix
charges and respect only the two parity blocks): the charge-zero sectors spanned by
$\{\ket{000},\ket{011}+\ket{101}+\ket{110}\}$ and $\{w,v\}$, and the one-dimensional sectors $e,\bar e,o,\bar o$.
The following observations come from floating-point searches and are not certified. Extreme points differ in how their
rank is distributed among these sectors. The surface of Section~\ref{sec:ppt3-surface}
occupies $u_s$, the whole $\{w,v\}$ sector, $e$ and $o$, while the optimiser behind $\tilde\rho$ occupies $u_s$, one
direction $w+cv$, $e$, $\bar e$ and $o$, with $s\approx0.974$ and $c\approx-0.750$: the same symmetries as the surface,
but one combination of $w$ and $v$ is traded for the opposite-chirality vector $\bar e$. Both have profile $(5,5,5,5)$.
The level-two white-noise tolerance of $\rho_{\rm opt}$ before mixing is numerically $2.36\times10^{-2}$; whether some fully-PPT
three-qubit state is more robust is open.
\end{remark}