Devising local protocols for multipartite quantum measurements

We provide a method of designing protocols for implementing multipartite quantum measurements when the parties are restricted to local operations and classical communication (LOCC). For each finite integer number of rounds $r$, the method succeeds in every case for which an $r$-round protocol exists for the measurement under consideration, and failure of the method has the immediate implication that the measurement under consideration cannot be implemented by LOCC no matter how many rounds of communication are allowed, including when the number of rounds is allowed to be infinite. It turns out that this method shows—often with relative ease—the impossibility by LOCC for a number of examples, including cases where this was not previously known, as well as the example that demonstrated what has famously become known as nonlocality without entanglement.

Figure 1

Summary of the results of this Appendix, depicting an LOCC protocol for the measurement of Eq. (A1). [Note that only the local operators are indicated next to each node in this diagram, rather than the full multipartite operator.]