Quantum Nonlocality Does Not Imply Entanglement Distillability

Entanglement and nonlocality are both fundamental aspects of quantum theory, and play a prominent role in quantum information science. The exact relation between entanglement and nonlocality is, however, still poorly understood. Here we make progress in this direction by showing that, contrary to what previous work suggested, quantum nonlocality does not imply entanglement distillability. Specifically, we present analytically a 3-qubit entangled state that is separable along any bipartition. This implies that no bipartite entanglement can be distilled from this state, which is thus fully bound entangled. Then we show that this state nevertheless violates a Bell inequality. Our result also disproves the multipartite version of a long-standing conjecture made by Peres.

Figure 1

Properties of the 3-qubit state ${\rho }_{\mathrm{Bell}}$. (i) Alice, Bob, and Charlie share ${\rho }_{\mathrm{Bell}}$ which is fully biseparable. Thus, no pure bipartite entanglement can be distilled on any bipartition. (ii) It turns out, however, that ${\rho }_{\mathrm{Bell}}$ can nevertheless violate a Bell inequality when judiciously chosen local measurements are performed. The joint probability distribution obtained in this way, i.e., $p_{{\rho }_{\mathrm{Bell}}}(abc|xyz)$, is nonlocal. This shows that quantum nonlocality does not imply entanglement distillability.