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.