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Let's assume a photon is moving in the $z$ direction and state of the photon is represented by $$|\psi\rangle = \alpha |x\rangle + \beta |y\rangle $$

This photon will pass through three polarizers. One is oriented in the y direction along the horizontal, second is by 45 degree and third one is along the $x$ direction.

I have written the matrix representation of the photon $$\begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix}$$

And let represent the state along $x$ and $y$ axis by $$\begin{pmatrix} 1 \\ 0 \end{pmatrix} \hspace{0.3cm}\text{and} \hspace{0.3cm}\begin{pmatrix} 0 \\ 1 \end{pmatrix}$$.

I understand what's the probability of getting each state ofter entering the each polarizer. My problem is that, i want to write the polarization state as a total Eigen ket space. Could you please enlighten me how to write the ket space whenever a polarized light enters into a polarizer. You can give me an example. I'm fine with that.

You are welcome to improve the question or ask me so I can add more information to the question.

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You probably just need to revisit the fundamentals of quantum mechanics. A polarizer is effectively a measurement device. For example the horizontal polarizer would correspond to a measurement operator $|y\rangle \langle y|$ in your notation, the vertical one, to $|x\rangle \langle x|$ and the diagonal to $(|x\rangle +|y\rangle )(\langle x|+\langle y|)/2$. The probability to pass a horizontal polarizer would be |$\langle y | \psi \rangle |^2$ and if the photon passes it, its state $becomes$ $| y \rangle$. The same logic applies for the rest of the polarizers.

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