  Quantum entanglement plays a central role in many areas of physics, from
quantum information science to many-body systems. In order to grasp the essence
of this phenomenon, it is fundamental to understand how different
manifestations of entanglement relate to each other. In 1999, Peres conjectured
that Bell nonlocality is equivalent to distillability of entanglement. The
intuition of Peres was that the non-classicality of an entangled state, as
witnessed via Bell inequality violation, implies that pure entanglement can be
distilled from this state, hence making it useful for most quantum information
protocols. Subsequently, the Peres conjecture was shown to hold true in several
specific cases, and became a central open question in quantum information
theory. Here we disprove the Peres conjecture by showing that an undistillable
bipartite entangled state---a bound entangled state---can nevertheless violate
a Bell inequality. This shows that Bell nonlocality implies neither
entanglement distillability, nor non-positivity under partial transposition,
thus clarifying the relation between three fundamental aspects of entanglement.
Finally, our results lead to a novel application of bound entanglement for
device-independent randomness certification.
