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(1) How to deterministically distinguish the following quantum states: $$\frac{1}{\sqrt{2}}[|+0\rangle|0\rangle+|-1\rangle|1\rangle$$, $$\frac{1}{\sqrt{2}}|-0\rangle|0\rangle+|+1\rangle|1\rangle$$, $$\frac{1}{\sqrt{2}}|-0\rangle|0\rangle-|+1\rangle|1\rangle$$, $$\frac{1}{\sqrt{2}}|+0\rangle|0\rangle-|-1\rangle|1\rangle$$ where $$|\pm\rangle=\frac{1}{\sqrt{2}}[|0\rangle\pm|1\rangle$$

Here the first two particles are with Alice and the third with Bob.

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    $\begingroup$ Hi NKaushik, and welcome to Physics Stack Exchange! We prefer questions to be conceptual and to demonstrate some of what the questioner has done to try to solve them. Accordingly, have you looked up anything about methods of distinguishing quantum states, or about the limits on one's ability to do so? Did you try applying them to your situation? What conceptual problem prevented you from being successful with that? All this would be good information to edit into the question. $\endgroup$
    – David Z
    Commented May 6, 2012 at 5:13
  • $\begingroup$ Here if Alice/Bob measures then the entangled character of the quantum system is lost and it is reduced to a product state. $\endgroup$
    – NKaushik
    Commented May 6, 2012 at 5:32

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If you want a deterministic local-operations-and-classical-communication protocol that will unambiguously determine which of the four states is present, it isn't possible. Alice can perform a degenerate measurement that collapses onto the spans of $|+0\rangle,|-1\rangle$ and $|-0\rangle,+1\rangle$, but after that the question is equivalent to determining under LOCC which of the Bell states $$|\phi^\pm\rangle_{AB}=\frac{1}{\sqrt{2}} \left[ |0\rangle_A |0\rangle_B \pm |1\rangle_A |1\rangle_B \right]$$ is present, and this is impossible - the best you can hope for using only classical communication is beating the CHSH Bell inequality.

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