Separability and entanglement in tripartite states

While classical correlations can be freely distributed among many systems, this is not true for entanglement and quantum correlations. If a quantum system S-a is entangled with another quantum system S-b, then its entanglement with any third quantum system S-c cannot be arbitrary. This is the celebrated monogamy of entanglement. Implicit in this general statement is the plausible belief that only entanglement between the systems S-a and S-b constrains the entanglement between S-a and the third system S-c. We demonstrate that even classical correlations between S-a and S-b may impose surprisingly stringent restrictions on the possible entanglement between S-a and S-c. In particular, perfect bipartite classical correlations and full entanglement cannot coexist in any tripartite state. An intuitive explanation of this monogamy of hybrid classical and quantum correlations might be that the system S-a has a correlating capability, which cannot be used to establish any entanglement with a third system (but can still be used to establish classical correlations) if it is exhausted when correlated with S-b (in either a classical or quantum fashion). This may be interpreted as an alternate version of monogamy.