# Quantum state of entangled photons

## Homework Statement

Suppose two polarization-entangled photons A and B in the following Bell state:

\begin{equation}
\Phi=\frac{1}{\sqrt{2}}\bigl(\left|H_{A},H_{B}\right\rangle + \left| V_{A},V_{B}\right\rangle\bigr)
\end{equation}

1. What is the state if the photon A passes through a double-slit?

2. What is the state if the photon A passes through a double-slit and the photon B passes through a linear polarizer oriented at the +45 angle?

2. The attempt at a solution

My attempt to solve (1):

A double-slit in the photon A's path creates a phase shift $$e^{i\Delta\phi}$$ because of unequal paths from each slit to a particular place on a screen. As the photon A can now take a path through one or the other slit, and there is a phase shift between the two,

\begin{equation}
\left|H_{A},H_{B}\right\rangle \longrightarrow \frac{1}{\sqrt{2}}\bigl(\left|H_{A},H_{B}\right\rangle\bigr) + e^{i\Delta\phi}\frac{1}{\sqrt{2}}\bigl(\left|H_{A},H_{B}\right\rangle\bigr)
\end{equation}

Similarly:

\begin{equation}
\left|V_{A},V_{B}\right\rangle \longrightarrow \frac{1}{\sqrt{2}}\bigl(\left|V_{A},V_{B}\right\rangle\bigr) + e^{i\Delta\phi}\frac{1}{\sqrt{2}}\bigl(\left|V_{A},V_{B}\right\rangle\bigr)
\end{equation}

Therefore:

\begin{equation}
\Phi=\frac{1}{2}\bigl(\bigl(1+e^{i\Delta\phi}\bigr)\left|H_{A},H_{B}\right\rangle + \bigl(1+e^{i\Delta\phi}\bigr)\left|V_{A},V_{B}\right\rangle \bigr)
\end{equation}

My attempt to solve (2):

The photon B passing through a 45-degree polarizer has a 50% probability of being absorbed and a 50% probability of converting:

\begin{equation}
H_{B} \longrightarrow \frac{1}{\sqrt{2}}\left|+45_{B}\right\rangle = \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr)
\end{equation}

\begin{equation}
V_{B} \longrightarrow \frac{1}{\sqrt{2}}\left|+45_{B}\right\rangle = \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr)
\end{equation}

Putting (1) and (2) together:

\begin{equation}
\Phi=\frac{1}{2}\bigl(\bigl(1+e^{i\Delta\phi}\bigr)\left|H_{A}\right\rangle \bigotimes \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr) + \bigl(1+e^{i\Delta\phi}\bigr)\left|V_{A}\right\rangle \bigotimes \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr) \bigr)
\end{equation}

The rest is easy, just manual work, but I'm wondering if I made any mistakes up to this point. Anyone willing to help?