Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 5 / a

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Solution

Use the ordered polarization basis

(∣ H ⟩, ∣ V ⟩)

and define the Bell states

∣ Φ^{\pm} ⟩ = \frac{∣ HH ⟩ \pm ∣ VV ⟩}{2}, ∣ Ψ^{\pm} ⟩ = \frac{∣ H V ⟩ \pm ∣ V H ⟩}{2} .

Regrouping the four-photon quantum state by channel pairs

(1, 4)

and

(2, 3)

gives

∣ Φ^{+} ⟩_{12} ∣ Φ^{+} ⟩_{34} = \frac{1}{2} (∣ Φ^{+} ⟩_{14} ∣ Φ^{+} ⟩_{23} + ∣ Φ^{-} ⟩_{14} ∣ Φ^{-} ⟩_{23} + ∣ Ψ^{+} ⟩_{14} ∣ Ψ^{+} ⟩_{23} + ∣ Ψ^{-} ⟩_{14} ∣ Ψ^{-} ⟩_{23}) .

For instance, projecting channels 2 and 3 on

⟨ Ψ^{-} ∣

retains the two original terms with inner polarizations

H V

and

V H

, with opposite signs, leaving

(∣ H V ⟩ - ∣ V H ⟩)_{14} / (22)

. Its squared norm is

1/4

.

By the same orthonormality argument, each Bell-basis measurement outcome prepares the identically named Bell state in channels 1 and 4, with

P (Φ^{+}) = P (Φ^{-}) = P (Ψ^{+}) = P (Ψ^{-}) = \frac{1}{4} .

Before communication of the outcome, Alice and Bob's density matrix is

I_{4} /4

; the usable conditional entanglement is established by the announced Bell-basis measurement result. This assumes an ideal complete Bell-basis measurement; ordinary passive linear optics alone does not deterministically distinguish all four polarization Bell states.

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