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

Use the ordered polarization basis $(|H⟩,|V⟩)$ and define the Bell statesRegrouping the four-photon quantum state by channel pairs $(1,4)$ and $(2,3)$ givesFor instance, projecting channels 2 and 3 on $⟨{\Psi }^{-}|$ retains the two original terms with inner polarizations $HV$ and $VH$, with opposite signs, leaving $(|HV⟩-|VH⟩{)}_{14}/(2\sqrt{2})$. 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, withBefore 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.

On Bob's polarization qubit, takeSuitable conditional unitary operators areIndeed $(I\otimes XZ)|{\Phi }^{+}⟩=|{\Psi }^{-}⟩$, $(I\otimes X)|{\Phi }^{-}⟩=|{\Psi }^{-}⟩$, and $(I\otimes Z)|{\Psi }^{+}⟩=-|{\Psi }^{-}⟩$. The last minus sign is a harmless global phase.

For an ideal half-wave plate whose fast axis makes angle $\theta$ with horizontal polarization, discard the common phase and use the Jones matrixIt implements $Z$ at $0^{\circ }$ and $X$ at $45^{\circ }$. Thus for outcome ${\Phi }^{+}$ use two half-wave plates, first $0^{\circ }$ then $45^{\circ }$ in the propagation direction; for ${\Phi }^{-}$ use one at $45^{\circ }$; for ${\Psi }^{+}$ one at $0^{\circ }$; for ${\Psi }^{-}$ use no plate. Ideal retardance is assumed at the photon wavelength, and the setting is chosen after receiving the detector's classical result.

Each source has density matrixAssume the two sources and the detector's random choices are independent. In the $p^2$ branch both resources are ideal. If either resource is $I_4/4$, the remote pair remains $I_4/4$ after any specified Bell-basis measurement outcome. An ideal detector and the correction of part (b) therefore yieldWhen the detector instead reports an independent uniformly random label, the corresponding four Pauli gates average the conditional pair to $I_4/4$. ThusHere $q$ is the probability of the reliable-detector branch, not its total chance of being correct: that chance is $q+(1-q)/4$.

For the usual two-setting CHSH inequality, the spin singlet state has correlation $-\mathbf{a}\cdot \mathbf{b}$ and the white-noise part has zero correlation, so $E(\mathbf{a},\mathbf{b})=-v\mathbf{a}\cdot \mathbf{b}$. With Alice's axes $z,x$ and Bob's axes $(z+x)/\sqrt{2},(z-x)/\sqrt{2}$, the absolute CHSH inequality combination is $2\sqrt{2}v$. These settings are optimal for an isotropic correlation tensor, givingThis is the threshold for the standard CHSH inequality with optimized projective polarization measurements. The unqualified phrase “the Bell inequality” needs this choice of test; entanglement alone is a weaker requirement, and no claim about every possible Bell inequality is needed for this calculation.

After the first Bell-basis measurement and the announced correction, the shared Werner state has visibility $v=p^{2}q$. Ideal quantum teleportation through that resource is the depolarizing channelOne way to see this is to separate the singlet branch, which teleports perfectly with the appropriate known correction convention, from $I_4/4$, which has no correlations and yields $I_2/2$.

The additional Bell-basis measurement at Alice's station has its own independent reliable branch of probability $q$. In its random-label branch the correction averages the four Pauli gates, again giving $I_2/2$. Hence the actual teleported density matrix isOn sifted BB84 rounds, Bob measures in Alice's preparation basis. The preserved component gives the correct bit with certainty; the maximally mixed component gives an error with probability $1/2$. ThereforeFor $p=q=1$ it vanishes; if either parameter is zero it is $1/2$. The result presumes successful full Bell-basis measurement outcomes and independent stated imperfections, without extra loss, polarization drift or dark counts.