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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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  • $\begingroup$ Cross-posted on QCSE. $\endgroup$ – S.D. Apr 2 '19 at 20:53
  • $\begingroup$ What do you mean by "separable"? $\endgroup$ – Norbert Schuch Apr 2 '19 at 23:37
  • $\begingroup$ Aren't separable POVMs more powerful than LOCC ones? $\endgroup$ – Norbert Schuch Apr 3 '19 at 10:03
  • $\begingroup$ Yes indeed, I made a bad mistake in formulating the question. I want a separable operation that is non LOCC. I corrected the mistake. I just have never seen a counterexample to all separables are LOCC. $\endgroup$ – MrRobot Apr 3 '19 at 10:14
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    $\begingroup$ MrRobot, edits which change the meaning of a post --- especially edits which invalidate existing answers --- are discouraged. I've tried to preserve both your original question and your intended meaning. $\endgroup$ – rob Apr 3 '19 at 20:13
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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).

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