What is an example of LOCC measurement which is not separable?

Question

Could you give me an example of a measurement which is LOCC (Local Operations Classical Communication) but not separable? Or better, one which is separable but not LOCC?

Given an ensable of states $\rho^{N}$, a separable measurement on it is a POVM $\lbrace N_i \rbrace$ where the effects $N_i$ are all of the form $N_i = A_i^{1} \otimes A_i^{2} \otimes \dots \otimes A_i^{N}$. So they are a separable product of effects acting on each state $\rho$ in $\rho^{N}$.

Is every separable measurement LOCC?

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Cross-posted on quantumcomputing.SE

Answer 1

There isn't: Any LOCC measurement is also a separable measurement. This is easy to see: Alice's first measurement has POVM elements $A_{i_1}\otimes I$. Alice then communicates her outcome $i_1$ to Bob. Bob's subsequent measurement has elements $I\otimes A^{i_1}_{i_2}$, where $i_2$ enumerates Bob's outcomes, and $A^{i_1}$ indicates that Bob's POVM can depend on Alice's outcome. The total POVM of both has then elements $$ N_{i_1,i_2}=A_{i_1}\otimes B^{i_1}_{i_2}\ , $$ which is a separable POVM with double index ${i_1,i_2}$. Clearly, this can be iterated to an arbitrary number of rounds, and generalized to an arbitrary number of parties, and will always have POVM elements of the form $N_i=A_i\otimes B_i\otimes \cdots$.

Conversely, not every separable POVM can be written as a LOCC POVM. A counterexample is given in Bennett et al., Quantum Nonlocality without Entanglement, Phys. Rev. A. 59, 1070 (1999).