quant-ph — Quantum Physics
Genuinely multipartite bound entanglement of three qubits
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 -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 white noise, while makes it biseparable. We also construct extreme examples with the smallest possible partial-transpose ranks . All our explicit states, including the full-rank one, lie in the mixed 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.