Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 7 / c / Solution

Use the same protocol: distribute one resource photon through $\mathcal{C}$, and send Alice's two classical bits error-free in the $D,A$ alphabet from part a. At $p=1/2$ the distributed resource isFor the second component Bob has an extra $X$ before his conditional correction. Conjugation by that correction changes $X$ at most by a sign, which cancels in the density operator. This bit-flipped Bell resource teleportation therefore induces the bit-flip map,Using the squared pure-target quantum fidelity convention,The maximum is $1$ precisely for the Pauli X eigenstates $D,A$. The minimum is $1/2$ whenever $\mathrm{Re}({\alpha }^{\ast }\beta )=0$, the great circle with zero $x$ component of the Bloch vector, including $H,V$ and both circular polarizations. Uniform averaging over pure qubits gives $\overline{F}=2/3$.

The resource-distribution and classical-encoding assumptions are essential. If a perfect shared pair had already been supplied independently and this channel were used only for the robustly encoded classical bits, teleportation would remain perfect even at $p=1/2$. Conversely, unprotected $H,V$ transmission of Alice's correction bits would add classical errors. The stated expression applies to the resource sent through the given channel and the error-free classical encoding explicitly available in part a.