# LOCC vs separable measurement

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?

• Cross-posted on QCSE. – S.D. Apr 2 '19 at 20:53
• What do you mean by "separable"? – Norbert Schuch Apr 2 '19 at 23:37
• Aren't separable POVMs more powerful than LOCC ones? – Norbert Schuch Apr 3 '19 at 10:03
• 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. – MrRobot Apr 3 '19 at 10:14
• 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. – rob Apr 3 '19 at 20:13

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 $$N_i=A_i\otimes B_i\otimes \cdots$$.