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I'm trying get a better understanding of Bell's inequality in the form

$$\left|E\left(\bf{a},\bf{b}\right) -E\left(\bf{a},\bf{c}\right)\right|\leq 1+E\left(\bf{b},\bf{c}\right)\enspace.$$

I'm considering the Bell state

$$\left|\psi\right\rangle= \frac{1}{\sqrt{2}}\left(\left|+\right\rangle_1\left|-\right\rangle_2- \left|-\right\rangle_1\left|+\right\rangle_2\right)\enspace.$$

and the expectation value ##E^{qt}## of the product of the result of a measurement of spin 1 in direction ##\bf{a}## and spin 2 in direction ##\bf{b}##

$$E^{qt}\left(\bf{a},\bf{b}\right)= \left\langle\psi\right|\vec{\sigma_1} \cdot\bf{a}\otimes \vec{\sigma_2} \cdot\bf{b} \left|\psi\right\rangle=-\bf{a}\cdot\bf{b}$$

which is a straigthforward calculation. I'm using the notation from 1.5.1 in

http://www.uibk.ac.at/exphys/photonik/people/gwdiss.pdf

which is, however, in German.

We can violate the inequality by choosing e.g. ##\bf{a}=e_x##, ##\bf{b}=\left(e_x+e_z\right)/\sqrt{2}##, ##\bf{c}=e_z## which yields

$$\left|-\frac{1}{\sqrt{2}}-0\right|=\frac{1}{\sqrt{2}}\leq 1-\frac{1}{\sqrt{2}}$$

which is obviously wrong. I think this choice of vectors also maximally violates the inequality.

However, if we choose ##\bf{a}=e_x##, ##\bf{b}=e_y##, ##\bf{c}=e_z##, then we get

$$\left|0-0\right|\leq 1-0$$

so the inequality is valid. I now wanted to find out exactly when the inequality breaks down and interpolated by choosing ##\bf{a}=e_x##, ##\bf{b}=\cos{\phi}\cdot e_y+\sin{\phi}\cdot \left(e_x+e_z\right)/\sqrt{2}##, ##\bf{c}=e_z## with ##0\leq\phi\leq\pi/2##. Plugging this in we get

$$\left|-\frac{\sin{\phi}}{\sqrt{2}}-0\right|=\frac{\sin{\phi}}{\sqrt{2}}\leq 1-\frac{\sin{\phi}}{\sqrt{2}}$$

and solving for equality yields ##\phi=\pi/4##.

So far so good, since the Bell inequality is valid for ##0\leq\phi\leq\pi/4##, there should be a local-realistic description of the system for these values of ##\phi##. Hence there must be a separable, most probably mixed density operator that yields the same expectation values as ##\left|\psi\right\rangle## for these choices of measurement angles. How can I find this density operator? I'm interested in its structure, and how it breaks down when crossing the magic angle ##\phi=\pi/4## from below.

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# Analyzing Bell's inequality for different measurement angles

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