Bell states |\boldsymbol \Phi^- \rangle and |\boldsymbol \Phi^+ \rangle measurements
In this blog post, I extend my exploration of Bell state measurements of the previous post here to the |\boldsymbol \Phi^- \rangle and |\boldsymbol \Phi^+\rangle Bell states.
Let’s begin by considering the |\boldsymbol \Phi^-(\nu_\alpha, \nu_\beta)\rangle Bell state, defined as:
|\boldsymbol \Phi^-(\nu_\alpha, \nu_\beta)\rangle = \frac{1}{\sqrt{2}} \left( |\mathbf x_\alpha, \mathbf x_\beta\rangle - |\mathbf y_\alpha, \mathbf y_\beta\rangle \right)
Using the beam splitter relations for creation operators and performing the necessary algebraic manipulations, we find that the \boldsymbol \Phi^- state, after passing through the beam splitter, transforms into an output state proportional to:
|\boldsymbol {\Phi^-}(\nu_\gamma, \nu_\delta)\rangle \propto \frac{1}{2} \left[ |\mathbf x_\gamma, \mathbf x_\gamma\rangle - |\mathbf x_\delta, \mathbf x_\delta\rangle - |\mathbf y_\gamma, \mathbf y_\gamma\rangle + |\mathbf y_\delta, \mathbf y_\delta\rangle \right]
This result indicates that for an input |\boldsymbol \Phi^-\rangle state, we expect to detect two photons in the same output channel (\gamma or \delta), with the possible combinations of polarizations being (\mathbf x, \mathbf x) in channel \gamma, (\mathbf x, \mathbf x) in channel \delta, (\mathbf y, \mathbf y) in channel \gamma, or (\mathbf y, \mathbf y) in channel \delta.
Now, let’s turn our attention to the fourth Bell state, | \boldsymbol \Phi^+(\nu_\alpha, \nu_\beta)\rangle:
|\boldsymbol \Phi^+(\nu_\alpha, \nu_\beta)\rangle = \frac{1}{\sqrt{2}} \left( |\mathbf x_\alpha, \mathbf x_\beta\rangle + |\mathbf y_\alpha, \mathbf y_\beta\rangle \right)
Following a similar procedure, we find that the |\boldsymbol \Phi^+\rangle state, after passing through the beam splitter, becomes proportional to:
|\boldsymbol {\Phi^+}(\nu_\gamma, \nu_\delta)\rangle \propto \frac{1}{2} \left[ |\mathbf x_\gamma, \mathbf x_\gamma\rangle + |\mathbf y_\gamma, \mathbf y_\gamma\rangle - |\mathbf x_\delta, \mathbf x_\delta\rangle - |\mathbf y_\delta, \mathbf y_\delta\rangle \right]
The output state for |\boldsymbol \Phi^+\rangle also involves detecting two photons in the same channel. The possible detection events include (\mathbf x, \mathbf x) in channel \gamma, (\mathbf y, \mathbf y) in channel \gamma, (\mathbf x, \mathbf x) in channel \delta, or (\mathbf y, \mathbf y) in channel \delta.
Comparing the output states for |\boldsymbol \Phi^-\rangle and |\boldsymbol \Phi^+\rangle with those for |\boldsymbol \Psi^-\rangle and |\boldsymbol \Psi^+\rangle from my previous post, we observe that each Bell state leads to a distinct pattern of photon detections in the output channels and polarization detectors. This demonstrates how even a simple beam splitter setup can provide partial information about the Bell state of input photons.
For more insights into this topic, you can find the details here.