State Amplitudes and Probabilities

Let’s now assume we have the equipment (the detectors in the figure below) needed to detect (and count) the number of photons in each beam (during a specified period of time). The equipment to detect individual photons is readily available in most college and university physics departments. Unfortunately, the equipment is too expensive to buy for yourself.

We will go through this process in some detail because it gives us a model we can use for getting information about the quantum state of other qubits. Using the setup illustrated in the figure below, Alice prepares a beam of light in a linearly polarized state using the first polarizing sheet but she doesn’t tell me the transmission axis direction. Our task is to find the polarization state of Alice’s beam using the output of the detectors in the figure below relative to the PBS vertical and horizontal polarization directions. That is, we want to find the numerical values of the state amplitudes in $\ket{S} = a_{\text{h}}\ket{\text{hlp}} + a_{\text{v}}\ket{\text{vlp}}$.