% Genuinely multipartite bound entanglement of three qubits.
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\hypersetup{pdfauthor={Antony Lewis},pdftitle={Genuinely multipartite bound entanglement of three qubits}}
\title{Genuinely multipartite bound entanglement\\of three qubits}
\newcommand{\Sussex}{Department of Physics \& Astronomy, University of Sussex, Brighton BN1 9QH, UK}
\author{\href{https://orcid.org/0000-0001-5927-6667}{Antony Lewis}\\[2pt]{\small\normalfont \Sussex}}
\date{}
\begin{document}
\maketitle
\begin{abstract}
If a three-party state is positive under partial transposition (PPT) across each of its three bipartitions, no local
protocol can distil any pure entanglement from it: neither singlets between any two parties nor GHZ states. All of its
entanglement is bound. We show that such a state can nevertheless be genuinely multipartite entangled (GME), meaning
that it cannot be prepared by mixing states in each of which some party is unentangled from the rest. This already
happens for three qubits, the smallest system in which the question arises and a common experimental setting
for multipartite entanglement; to our knowledge no three-qubit example was known. An explicit mixture of five orthogonal
states (one GHZ-type, three $W$-type and one product state) is PPT across every cut and GME throughout an exact parameter interval, and its rank, five, is
the minimum possible. Genuine entanglement is proved by a witness acting only on the five-dimensional support, whose
nonnegativity on the biseparable states in that support reduces to a sum of squares on its curves of product vectors. Such
states are invisible to the PPT-mixture test, the standard semidefinite relaxation used to certify GME, yet they form
a set with nonempty interior among density matrices: an explicit full-rank example remains GME under $2.33\%$ white noise, while $2.336\%$
makes it biseparable. We also construct extreme examples with the smallest possible partial-transpose ranks
$(5,5,5,5)$. All our explicit states, including the full-rank one, lie in the mixed $W$ class: they are mixtures of pure
states with vanishing three-tangle, so their genuine entanglement needs no GHZ-type component. Full PPT therefore does not imply
biseparability, even for three qubits.
\end{abstract}

\section{Introduction}\label{sec:ppt3-intro}

Entanglement among three or more parties comes in two kinds. A state is \emph{biseparable} if it is a mixture of states
each separable across one of the bipartitions $A|BC$, $B|AC$, $C|AB$; the bipartition may differ from term to term. Otherwise it is \emph{genuinely multipartite entangled} (GME). Each term of a biseparable mixture is entangled at most within one pair of
parties, so GME is the multipartite property that experiments with several qubits usually aim to
certify~\cite{HHH09,GT09}. Showing that a mixed state is GME is harder than showing it entangled across
each cut: a state can be entangled across every bipartition and still be a mixture of states separable across
different ones. Biseparability is a property of the convex hull of three sets of separable states, not of each set
separately.

Entanglement can also be distillable or bound. Entanglement across a cut is \emph{distillable} if many copies can
be converted to singlets across that cut by local operations and classical communication (LOCC), and \emph{bound}
otherwise. A state that is positive under partial transposition (PPT) across a cut is undistillable across it~\cite{HHH98},
and PPT is also necessary for separability~\cite{Peres,HHH96}. For three qubits $A$, $B$, $C$ we call a state
\emph{fully PPT} when its partial transposes across $A|BC$, $B|AC$ and $C|AB$ are all positive. Every LOCC protocol among
the three parties is in particular LOCC across each cut, and such operations preserve PPT across that cut~\cite{HHH98,HHH09}.
Consequently no pure state entangled across any cut, neither singlets between two parties nor GHZ states, can be
distilled from a fully-PPT state, even asymptotically.
Whatever entanglement a fully-PPT state has is bound.

This paper asks whether bound entanglement of this extreme kind can still be genuinely multipartite, and answers yes
for three qubits. A fully-PPT GME state carries entanglement that no single bipartition accounts for and from which no
pure entanglement can be distilled: it is present only collectively. The question also has a practical side. The standard
way to certify GME in experiments and numerics relaxes biseparability to \emph{PPT mixtures}, mixtures of states each PPT
across one cut, which gives a semidefinite program~\cite{JMG,NMG}. A fully-PPT state is already a one-term PPT mixture,
so its genuine entanglement is invisible to this test and to any test built from PPT conditions across cuts and their
mixtures. Novo, Moroder and G\"uhne conjectured that for three qubits PPT mixtures and biseparable states coincide~\cite{NMG};
Ha and Kye refuted this with a state PPT across one chosen cut~\cite{HK}. A fully-PPT GME state is the strongest form of
the counterexample.

Three qubits are where such a state is least expected. Each cut is $2\times4$, just beyond the $2\times2$ and $2\times3$
systems where PPT and separability coincide~\cite{HHH96}, and low rank forces separability: a fully-PPT three-qubit
state of rank at most four is separable across a cut, hence biseparable~\cite{KCKL,HLVC,KL,ABLS}. A fully-PPT GME state therefore
has rank at least five. Examples are known for more qubits and for higher local dimension (Section~\ref{sec:ppt3-prior}).
For three qubits, earlier constructions and numerical states were entangled but not shown to be GME, or were PPT across
one cut only~\cite{HK,KRB,GIM}, and Singh and Banik noted that no explicit example is known and derived necessary
rank conditions for one~\cite{SB}.

Our method follows the physics of the convex hull. Every pure state in a decomposition of $\rho$ lies in its support,
so a biseparable decomposition of a low-rank state can use only those vectors of the support that are products across some
cut. We choose a five-dimensional support $R$, spanned by five orthogonal vectors (one GHZ-type, three $W$-type and $\ket{111}$), in which the cut-product
vectors form one explicit curve per cut, parametrised by a single complex number and related by cyclic permutation of the
qubits. A Hermitian form that is nonnegative on these three curves is nonnegative on every biseparable state supported on
$R$; its negative expectation in our state then proves GME. For the form we use, nonnegativity on the curves is an explicit
factorisation into squares. Full PPT is a separate and elementary step, since cyclic and parity symmetry reduce each
partial transpose to two $4\times4$ blocks. Robustness to noise needs a test stronger than PPT; we use exact rational
certificates from the second level of the symmetric-extension hierarchy, together with explicit biseparable decompositions
that bound the noise thresholds from above.

Our main results are the following.
\begin{enumerate}[label=(\arabic*),itemsep=0pt,topsep=2pt]
\item \emph{Minimal-rank family} (Theorem~\ref{thm:ppt3-family}). A mixture $\rho_t$ of five orthogonal
vectors (GHZ-type, $W$-type and $\ket{111}$) has rank five and is fully PPT and GME for $3.257\lesssim t\lesssim3.389$ (exact endpoints in the
theorem).
\item \emph{A witness that sees only the support} (Proposition~\ref{prop:ppt3-witness}). One diagonal Hermitian form,
nonnegative on the product curves by a square factorisation, excludes biseparable mixtures across cuts, with no
semidefinite-program solver.
\item \emph{Extreme states} (Sections~\ref{sec:ppt3-family} and~\ref{sec:ppt3-surface}). Extreme fully-PPT GME states have
partial-transpose rank profiles $(5,6,6,6)$ and, the smallest possible, $(5,5,5,5)$; the latter lie in an analytic family
that is fully PPT and GME on the whole window $1/\sqrt3<\tau<1$ (one parameter modulo local filters). Every state of the family is
extreme in the fully-PPT set, and near every member the extreme states of this profile form a $27$-dimensional manifold with
nine local-filter moduli.
\item \emph{One support, both phases} (Section~\ref{sec:ppt3-coexistence}). One rank-five support carries both
biseparable and GME fully-PPT states, separated by an exact transition inside the strictly PPT region, so the support alone
does not decide genuine entanglement.
\item \emph{Robustness} (Section~\ref{sec:ppt3-noise}). Fully-PPT GME states have positive volume. An explicit full-rank
fully-PPT state remains GME under $2.33\%$ white noise, $1\%$ independent depolarisation per qubit, or a trace-distance
perturbation of $7.3\times10^{-3}$; its white-noise threshold is known to within half a per cent in relative terms. A
surface state of the $(5,5,5,5)$ family tolerates about $0.43\%$.
\item \emph{Structure} (Sections~\ref{sec:ppt3-chirality} and~\ref{sec:ppt3-relaxations}). No fully-PPT GME state is invariant
under the exchange of two qubits, so none is permutation invariant~\cite{NMG}, and our family is not locally equivalent
to its complex conjugate, not even by invertible local filters. Every explicit state we construct, including both extreme states
and the full-rank state $\tilde\rho$, belongs to the mixed $W$ class (Theorem~\ref{thm:ppt3-w-class}), as conjectured for PPT entangled three-qubit states by Ac\'in et al.~\cite{ABLS}. For every rank-five fully-PPT GME state the product vectors of its support form three rational
quartic curves, so supported witnesses of the kind used here detect all of them
(Theorem~\ref{thm:ppt3-universal-moments}).
\end{enumerate}

Section~\ref{sec:ppt3-relaxations} places the results relative to PPT mixtures and earlier work. The main
text gives the proof strategy; the computations are in Appendices~\ref{sec:ppt3-app-family}, \ref{sec:ppt3-app-surface},
\ref{sec:ppt3-app-coexistence} and~\ref{sec:ppt3-app-noise}, in the order of the sections they serve. Appendices~\ref{sec:ppt3-app-moment}
and~\ref{sec:ppt3-kernel} prove the product-kernel lemma and the moment cone of Section~\ref{sec:ppt3-relaxations} and the kernel-penalty
lift of the supported witness, and Appendix~\ref{sec:ppt3-verifiers} lists the proof dependencies. All theorems,
propositions, lemmas and corollaries are proved, analytically or with the exact computations named next; numerical
observations appear only in remarks and are labelled as such. Exact rational or algebraic
computations are used only for the extremality of $\sigma_6$ and the independent extremality check on the $(5,5,5,5)$ surface, the moduli count, the phase-coexistence minors and the noise certificates. The verification scripts with their
captured output are in a companion repository~\cite{Companion}, together with a Lean~4 formalisation of the main analytic
results and of the extremality of one state of the surface; the noise certificates, the moduli count, the extremality of
$\sigma_6$ and of the whole surface, and the minimality of rank five are checked by the exact scripts or cited, not formalised.

\input{family}
\input{rank_five_surface}
\input{w_class_decompositions}
\input{phase_coexistence}
\input{noise_certificates}
\input{relaxations_and_scope}

\section{Discussion}\label{sec:ppt3-discussion}

Physically, the examples separate two questions that are easy to conflate: whether entanglement can be extracted across
some bipartition, and whether it is shared by all three parties. Even for three qubits, the first does not decide the second. A
fully-PPT GME state yields no singlet across any cut and no GHZ state under any LOCC protocol, yet it is not a mixture of
states each separable across some bipartition. The relevant gap is between the intersection of the three PPT sets and
the convex hull of the three cut-separable sets. The support of $\rho_t$ contains product vectors across every cut, but no convex mixture of them
reproduces $\rho_t$; the supported witness certifies this failure of mixing. Symmetry plays a definite role.
Reflection symmetry together with PPT forces separability across a cut (Corollary~\ref{cor:ppt3-symmetrisation}), so the
states must break every exchange of two qubits; ours do so through the cyclic phases of the $W$-type vectors, and they
are also locally chiral (Corollary~\ref{cor:ppt3-chirality}).

For certification, PPT-mixture tests cannot certify the genuine entanglement of these states; a stronger relaxation of
biseparability is one way to do so. The second level of the
symmetric-extension hierarchy suffices here, and for white noise its certified margins nearly match explicit biseparable
decompositions (Proposition~\ref{prop:ppt3-noise-sharp}), so the small size of the margins is a property of the states,
not of the relaxation. By
Proposition~\ref{prop:ppt3-noise-extremal} a most robust state can be taken extreme in the fully-PPT set, and numerical searches
suggest that within one symmetry class the extreme points differ in how their rank is distributed among the symmetry
sectors (Remark~\ref{rem:ppt3-robust-structure}). An NMR experiment has generated pseudo-bound-entangled fully-PPT three-qubit
states that are not GME~\cite{KPBS}. The state $\tilde\rho$ remains GME under depolarising errors of up to one per cent
per qubit, but generic preparation errors need much tighter control: its certified PPT-stability radius, $2.0\times10^{-5}$, is far
smaller than its GME radius, $7.3\times10^{-3}$ (Corollary~\ref{cor:ppt3-robust}); certifying such a state also requires witnesses that go beyond
partial transposition.

Several questions remain open: whether some fully-PPT three-qubit state tolerates substantially more noise, how to
describe the family of $\tilde\rho$ analytically, and how to decompose a witness into practical local measurements. The mixed-$W$ conjecture of
Ac\'in et al.~\cite{ABLS} holds for all our explicit states, including $\tilde\rho$ (Theorem~\ref{thm:ppt3-w-class}), but
remains open for general PPT entangled three-qubit states. A separate problem is whether every rank-five three-qubit state with maximally mixed one- and
two-qubit marginals is biseparable (Conjecture~\ref{conj:ppt3-flat}). Our rank-$(5,5,5,5)$ examples show that
the rank profile alone cannot prove it; the flat-marginal condition must play an essential role.

\section*{Use of AI}
All results, documents and scripts were generated by AI agents (Claude Opus and Sonnet 5.5, OpenAI's GPT 6.1 sol and Gemini 4 Argon/Carbon)
with human supervision and many cross-checks, including AI-generated Lean verification of the main analytic results.
I thank DeepMind and Anthropic for access to their scientist programs.

\section*{Acknowledgements}
We acknowledge support by the UK STFC grant ST/T000473/1.

\appendix
\small
\input{appendix_family}
\input{appendix_surface}
\input{appendix_coexistence}
\input{appendix_noise}
\input{appendix_moment_cone}
\input{kernel_penalty_lemma}
\input{verification_map}

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\end{document}