Is there a class of <PERSON> inequalities that are violated by certain mixtures of known orthogonal entangled states?
Is there a class of <PERSON> inequalities that are violated by certain mixtures of known orthogonal entangled states?
There exist a multitude of classes of <PERSON> inequalities that can be used in conjunction with the corresponding experimental designs.
<PERSON> inequalities from no-signalling distributions.
All the <PERSON> inequalities.
We will consider the following entangled states (Bell states):
$\Phi^+=\frac{1}{\sqrt{2}}(\vert 00\rangle+\vert 11\rangle)$
$\Phi^-=\frac{1}{\sqrt{2}}(\vert 00\rangle-\vert 11\rangle)$
$\Psi^+=\frac{1}{\sqrt{2}}(\vert 01\rangle+\vert 10\rangle)$
$\Psi^-=\frac{1}{\sqrt{2}}(\vert 01\rangle-\vert 10\rangle)$
<PERSON> and <PERSON> measure their photons using coincidence circuits and can perform a statistical analysis on the outcome of their measurements. They choose a class of <PERSON> inequalities to analyze as well as the corresponding experimental setup (the <PERSON> test experiment ) in order to assess whether the chosen inequalities are violated (in order to prove the presence of quantum entanglement ).
Case 1. The source sends <PERSON> and <PERSON> only entangled pairs in the state $\Phi^+$ . In this case it is known that there is a class of <PERSON> inequalities and <PERSON> test experiments that shows the violation of these inequalities. That goes back to <PERSON>'s work. Similar results are valid if the source sends only entangled pairs in any of the other <PERSON> states $\Phi^-$ , $\Psi^+$ , or $\Psi^-$ .
Case 2. The source sends <PERSON> and <PERSON> entangled pairs in the state $\Phi^+$ with probability $\alpha$, or in the state $\Phi^-$ with probability $\beta$ , or in the state $\Psi^+$ with probability $ \gamma$, or in the state $\Psi^-$ with probability $\delta$ , where $\alpha + \beta + \gamma + \delta = 1$. In this case is there a class of <PERSON> inequalities and <PERSON> test experiments that shows the violation of these inequalities ? Also for what range of the parameters $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ is this possible?
Question. 680
In other words, can <PERSON> and <PERSON> prove the presence of quantum entanglement (correlations ) when they receive a mixture of known orthogonal entangled states? And for what range of the parameters?
Clearly for $\alpha = 1$ and $\beta = \gamma = \delta = 0$ this is possible (this is case 1).
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Justification for this question. In the following I will describe the reason/motivation behind this question. This part is independent of the question itself but it is worth mentioning. It doesn't obscure the clarity of the question above.
Delayed choice entanglement swapping.
Two pairs of entangled photons are produced, and one photon from each pair is sent to a party called <PERSON>. Of the two remaining photons, one photon is sent to the party <PERSON> and one is sent to the party <PERSON>. <PERSON> can now choose between two kinds of measurements. If he decides to measure his two photons in a way such that they are forced to be in an entangled state, then also <PERSON>'s and <PERSON>'s photon pair becomes entangled.
If <PERSON> chooses to measure his particles individually, <PERSON>'s and <PERSON>'s photon pair ends up in a separable state. Modern quantum optics technology allows to delay <PERSON>'s choice and measurement with respect to the measurements which <PERSON> and <PERSON> perform on their photons. Whether <PERSON>'s and <PERSON>'s photons are entangled and show quantum correlations or are separable and show classical correlations can be decided after they have been measured.
We follow the calculations in the reference. 682