 Solution

Wave plates and polarization phase shifters implement deterministic single-qubit unitaries: a suitable quarter-wave/half-wave/quarter-wave sequence realizes an arbitrary polarization rotation. A deterministic entangling gate on two independently encoded photons needs an effective interaction or additional resources that make one photon's logical action conditional on the other. For example, a sufficiently strong conditional phase followed by Hadamards on the target implements CNOT gate. Ordinary passive linear optics alone does not provide this deterministic interaction on independent polarization qubits. The practical difficulty is that optical nonlinearities at the single-photon level are weak and commonly accompanied by loss or noise; auxiliary photons, measurements and feed-forward trade determinism for resource overhead.

The polarizing beam splitter eigenpolarizations at angle $\alpha$ are the orthonormal statesThe first is perfectly transmitted and the second perfectly reflected, from either input port. Port-dependent reflection phases can be fixed by a consistent optical phase convention or compensated locally; they do not alter the transmitted/reflected basis.