Single-photon gates for Bell state manipulation
It is still impossible at the present time to have an apparatus that can measure all the Bell states at the same time. However, we can manipulate them using tools like birefringent plates, which act as single-photon gates. These gates can perform transformations on the polarization of individual photons, effectively acting as qubit gates.
Consider a general polarization state of a photon:
|\boldsymbol \varphi(\nu)\rangle = \alpha | \mathbf x \rangle + \beta | \mathbf y \rangle
A birefringent plate can apply a unitary transformation \mathbf M:
\begin{bmatrix} \alpha^\prime \\ \beta^\prime \end{bmatrix} = \mathbf M \begin{bmatrix} \alpha \\ \beta \end{bmatrix}
Two important gates are the Pauli \mathbf X-gate, represented by the matrix:
\mathbf X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
and the phase gate, represented by:
\mathbf S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}
Starting with the Bell state |\boldsymbol \Phi^+(\nu_1, \nu_2)\rangle, we can apply a Pauli \mathbf X-gate to photon \nu_2 to obtain |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle:
(\mathbf I \otimes \mathbf X_2) |\boldsymbol \Phi^+(\nu_1, \nu_2)\rangle = |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle
Similarly, by applying the phase gate \mathbf S twice, we can transform |\boldsymbol \Phi^+(\nu_1, \nu_2)\rangle into |\boldsymbol \Phi^-(\nu_1, \nu_2)\rangle. This can be achieved by applying \mathbf S^2 to either photon \nu_1, photon \nu_2, or applying \mathbf S to both photons:
\begin{aligned} (\mathbf S_1^2 \otimes \mathbf I_2) |\boldsymbol \Phi^+(\nu_1, \nu_2)\rangle & = |\boldsymbol \Phi^-(\nu_1, \nu_2)\rangle \\ (\mathbf I_1 \otimes \mathbf S_2^2) |\boldsymbol \Phi^+(\nu_1, \nu_2)\rangle & = |\boldsymbol \Phi^-(\nu_1, \nu_2)\rangle \\ (\mathbf S_1 \otimes \mathbf S_2) |\boldsymbol \Phi^+(\nu_1, \nu_2)\rangle & = |\boldsymbol \Phi^-(\nu_1, \nu_2)\rangle \end{aligned}
By combining these operations, we can generate all four Bell states from a single starting state, demonstrating the power of single-photon gates in quantum information processing.
For more insights into this topic, you can find the details here.