Entanglement, information, and multiparticle quantum operations

Collective operations on a network of spatially separated quantum systems can be carried out using local quantum operations, classical communication (CC), and shared entanglement (SE). Such operations can also be used to communicate classical information and establish entanglement between distant parties. We show how these facts lead to measures of the inseparability of quantum operations, and we argue that a maximally inseparable operation on two qubits is the SWAP operation. The generalization of our argument to N-qubit operations leads to the conclusion that permutation operations are maximally inseparable. For even N, we find the minimum SE and CC resources which are sufficient to perform an arbitrary collective operation. These minimum resources are $2\left(N-1\right)$ units of entanglement and $4\left(N-1\right)$ bits, and these limits can be attained using a simple teleportation-based protocol. We also obtain lower bounds on the minimum resources for the odd case. For all $N>~4,$ we show that the SE and CC resources required to perform an arbitrary operation are strictly greater than those that any operation can establish or communicate.