Family Of Bell Inequalities Violated By Higher-Dimensional Bound Entangled States

We construct (dxd)-dimensional bound entangled states, which violate, for any d > 2, a bipartite Bell inequality introduced in this paper. We conjecture that the proposed class of Bell inequalities acts as a dimension witness for bound entangled states: For any d > 2 there exists a Bell inequality from this class that can be violated with bound entangled states only if their Hilbert space dimension is at least dxd. Numerics supports this conjecture up to d = 8.