Could you give me an example of a measurement which is separable but not LOCC (Local Operations Classical Communication)? 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 $\rho^{N}$.

Note: Cross-posted on Physics SE.

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    $\begingroup$ Could you define what you mean by separable? For example, if Alice flips a coin and tells the result to Bob, and the coin flip determines whether they both do an X or a Z measurement, is that separable or not? $\endgroup$ Apr 2 '19 at 22:59
  • $\begingroup$ Yeah, that is separable for sure as it is LOCC. I made a bad mistake now fixed in the question. I want a separable measurement which is not LOCC, not the opposite. $\endgroup$
    – MrRobot
    Apr 3 '19 at 9:56
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    $\begingroup$ This paper arxiv.org/pdf/0810.2327.pdf (see page 14) defines LOCC norm and SEP norm. It seems that you are using the definition of SEP norm (i.e. set of measurements) in their paper. The LO norm here is only regarding fixed cut, but the SEP allowing all kinds of cut. It might be the answer you are looking for.. $\endgroup$
    – Yupan Liu
    Apr 5 '19 at 20:08

Such an example is given in Bennett et al., Quantum Nonlocality without Entanglement, Phys. Rev. A. 59, 1070 (1999).


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