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From Malus's law we know that if we measure a the polarisation of light with a filter angle $\theta$ to the direction of polarisation then the intensity goes like: $$I=I_0 \cos^2(\theta/2)$$ Firstly does this still hold in quantum physics?

Further we know that for two entangled spin-1/2 particles the correlation between measurements of their spin by detectors at angle $\theta$ to one another is: $$c(\theta)=-\cos(\theta)$$ My question is what is the correlation between the polarisation of two entangled photons when each is measured by a different detector at an angle $\theta$ to each other?

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Once you measure one of the entangled photons you will know the state of the other too.

For simplicity assume that the two photons are entangled in a way they have the same polarization angle.

Let's call the two photons as "left and "right" and also call the detectors like this on each side.

There are two possibilities:

  • The left photon passes the polarizer. This means it aligned to the same angle as the left polarizer. The right one must also align due to the entanglement. So the chance it passes the right polarizer is $\mathrm{cos}^2(\theta)$ where $\theta$ is the difference in the angles.

  • The left photon fails the polarizer. Then you know the left one must be aligned perpendicularly to polarizer to fail it, so does right one, in this case chance the right one passes the polarizer is $\mathrm{cos}^2(\theta + \pi/2) = \mathrm{sin}^2(\theta)$. The chance it also fails it is $(1-\mathrm{sin}^2(\theta)) = \mathrm{cos}^2(\theta)$.

These two possiblities are exclusive so their chances add:

$\mathrm{cos}^2(x)\mathrm{cos}^2(\theta) + (1-\mathrm{cos}^2(x))\mathrm{cos}^2(\theta)$. Where $x$ is any random variable.

After adding together the $x$ terms fall out so the chance of coincidence is: $\mathrm{cos}^2(\theta)$.

It's a value between 0, 1 to get correlations we must map it between -1 and 1. So the correleation in this case will be $2\mathrm{cos}^2(\theta) - 1$.

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  • $\begingroup$ Do you have a source/any references for your first bullet point? Also do you know if Malu's law holds in quantum physics? $\endgroup$ – Quantum spaghettification May 14 '15 at 18:06
  • $\begingroup$ @Joseph Search for 'quantum malus law'. Basically it's the same as the classical one. But instead of relating the intensity of light beams, it tells the chance whether a photon with a given angle pass or fail the polarizer. $\endgroup$ – Calmarius May 14 '15 at 21:07

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