Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 5

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.

Before the gates, write the two Bell states asThe CNOT gate sends the last bit to $b\oplus a$. Projection onto outcome 0 on qubit 4 therefore requires $b=a$. After the Hadamard gate on qubit 2, projection onto its outcome 0 contributes $1/\sqrt{2}$ for either remaining value of $a$. The unnormalized surviving state isThusThis is entanglement swapping: photons 1 and 3 become entangled even though the joint measurement acts only on their original partners.

For heralded Bell-pair projection with two polarization parity filters, a Hadamard gate followed by computational-basis detection is a measurement in the diagonal polarization basis $|\pm ⟩=(|H⟩\pm |V⟩)/\sqrt{2}$. It can therefore be realized directly by a polarizing beam splitter at $45^{\circ }$ with a photodetector in each output: its transmitted and reflected clicks label $+$ and $-$. No separate Hadamard gate plate is required for this measurement.

Conditioning on one photon in each output of the first polarizing beam splitter implements the even $Z$-parity filterThe $45^{\circ }$ polarizing beam splitter similarly implements the even $X$-parity filter ${\Pi }_X^{46}=(I+X_4{X}_6)/2$. These are ideal coherent filters, with the optical phases chosen consistently. The all-zero photodetector record projects the upper outputs onto $⟨++{|}_{25}$ and the lower outputs onto $⟨00{|}_{46}$. The two effective bras areTherefore the unnormalized retained state isFor example, the simultaneous Bell contractions require the three pair bits to coincide, leaving only the $000$ and $111$ terms of the original pair expansion. Each surviving $|00{⟩}_{13}$ or $|11{⟩}_{13}$ amplitude is $1/(8\sqrt{2})$. ConsequentlyThis probability is conditioned on production of all three input pairs, but not yet on successful polarizing beam splitter routing. The routing probability itself is $⟨{\Pi }_Z^{25}{\Pi }_X^{46}⟩=1/4$: each nonidentity expectation vanishes on the relevant initial reduced state. Thus, if the question instead regards the routing postselection as already performed,These two denominators describe different conditioning. Neither includes the probability that the three crystals actually emit the required pairs, photodetector inefficiency or undesired multipair events.

A herald tells the experimenter when an entangled resource has actually been prepared while leaving the two resource photons available for subsequent operations. This permits synchronization, conditional storage or switching, and feed-forward. Direct down-conversion emission is probabilistic; inferring that a pair existed by destructively detecting both photons does not leave that same pair available as an input resource.

The second network uses passive optical elements, auxiliary entanglement and detection rather than a strong deterministic photon–photon CNOT gate interaction. Its smaller success rate can be acceptable if trials are inexpensive and successful events are reliably announced. Heralding is still probabilistic, not deterministic on demand; losses, dark counts and multiple-pair emission can give false or degraded heralds. The ideal $1/64$ result assumes the specified input pairs and valid detection model, so it should not be confused with a complete laboratory production efficiency.