  We construct ($d\times d$)-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 $d\times d$. Numerics
supports this conjecture up to $d=8$.

Quantum Physics