# Can you project on an orthogonal basis for a multipartite system using only local measurements and classical communication?

Say Alice possesses one qubit, and Bob two, and that the joint state is $$|\psi_{A, B_1, B_2}\rangle = \alpha|n_1\rangle + \beta |n_2\rangle$$, where $$|n_1\rangle$$ and $$|n_2\rangle$$ are orthonormal basis states for the combined Hilbert space. If you have access to all qubits, there obviously is a measurement which projects on the basis that includes $$|n_1\rangle$$ and $$|n_2\rangle$$.

However, what if you only allow Alice and Bob to do local measurements on their qubits? Then, I assume that Bob would need to send at least one classical bit to Alice for a joint measurement to be possible. Do they also need local quantum registers? Can we say anything about when such a joint measurement is possible?

Perhaps a concrete example would be: Let's say Alice and Bob share the state $$|\psi_{A, B_1, B_2}\rangle = \alpha|n_1\rangle + \beta |n_2\rangle$$. Is it possible, using only LOCC, to end up in the joint state $$|n_1\rangle$$ with probability $$|\alpha|^2$$, and in $$|n_2\rangle$$ with probability $$|\beta|^2$$, so that afterwards both parties know what the shared state is?

I assume that the precise implementation of such a measurement will depend on the states $$|n_1\rangle$$ and $$|n_2\rangle$$, but is there a general rule for when such a joint measurement is possible?

• The answer is probably no if $|n_1\rangle$ and $|n_2\rangle$ are entangled states Commented Sep 30, 2023 at 16:40
• Yes, that might very well be. The counterexample offered here is also based on that : physics.stackexchange.com/questions/782508/… Commented Sep 30, 2023 at 16:47