Optimal quantum cloning of orbital angular momentum photon qubits through Hong–Ou–Mandel coalescence. E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi and E. Santamato in Nature Photon. 3:720 (2009).  What the paper says!?

Here, the authors

report the first observation of the Hong–Ou–Mandel coalescence of two incoming photons having non-zero OAM

They do this «by making use of the recently demonstrated spin-OAM information transfer tools» which is their previous work, but for now I'm more interested in the HOM on vortices.

They also tackle the question of (optimal) quantum cloning, which is more related because one technique is the "symmetrization technique" where:

the bosonic nature of photons (that is, the symmetry of their overall wavefunction) is used within a two-photon HOM coalescence effect.

So that's the two things they are doing:

1. first observation of HOM interference of non-zero OAM.
2. first demonstration of the 1 → 2 universal optimal quantum cloning (UOQC) of the OAM

the beamsplitter is not an OAM-preserving optical device, as the reflection on the beamsplitter inverts the sign of the OAM. Therefore, the maximal two-photon coalescence is expected to be observed when the input photons carry exactly opposite OAM.

For HG, however:

Moreover, coalescence is expected when they have the same HG-like state, for example, ∣h> or ∣v>.

(rather than «Moreover,» I believe that "In contrast" would have been more clear.)

This is clear and clean.

Then they turn to the cloning, implementing the known UOQC scheme to this spatial degree of freedom. Here one needs to delve into the theory, which is interesting, and well detailed in their paper. They succinctly overview the projector that result from sending an unknown qubit to clone $\ket{\varphi}_{o2}$ on one port, and the mixed density matrix $(\ketbra{+2}{+2}+\ketbra{-2}{-2})/2$ on the other port. Their interference, it is shown, results in a mixed superposition with both photons being in the same state ${5\over6}\ketbra{\varphi}{\varphi}_{o2}+{1\over6}\ketbra{\varphi^\perp}{\varphi^\perp}_{o2}$, so that the input state is degraded in purity but the other has probability $5/6=0.8\bar3$ to be its clone. The probability 5/6 is the bound (optimum) from the general optimum qubit cloning fidelity $F=\frac{2N+1}{3N}$ (equal 5/6 for N=2).

They ran the protocol for six states (spanning the full space) and report very close to optimum results. They don't clarify why they use $\ell=\pm2$ as opposed to $\pm1$. It should work the same.

They use Padgett and Courtial's mapping of OAM to the Bloch sphere, producing the nice "cloning" above.

On high-dimensionality entanglement:

The orbital angular momentum (OAM) of photons lies in an infinitely dimensional Hilbert space, so it is a natural choice for implementing single-photon qudits, the units of quantum information in a higher dimensional space.

Nicely put:

an OAM state of light can also be regarded as an elementary form of optical image, so that OAM manipulation is related to quantum image processing