Using a clever design of polarization optic, Italian researchers have successfully created fourlevel 'ququart' quantum states using the polarization and orbital angular momentum of single photons. This approach may help to realize more effective forms of quantum communication.
Enormous progress has been made in the field of quantum communication over the past two decades. What was merely a collection of beautiful ideas in the late 1980s has now developed into an interdisciplinary area of science with emerging technological applications. In quantum communication, information is usually encoded in qubits — quantum systems with two orthogonal states that are used to represent a binary digital bit of information (one or zero). However, there is increasing interest in exploiting protocols that use systems of higher dimensionality, which have a greater number of logic levels than binary systems. At a generic level, multilevel quantum systems are called qudits, with specific names given according to the number of levels; for example, qutrits for three levels and ququarts for four levels.
Qudits are exciting because of the advantages they offer for quantum information processing. For example, quantum key distribution — one of the most developed and commercially relevant applications of quantum information science — is more secure and noise resilient with quditbased encoding. Furthermore, qudits may also aid in the realization of more efficient quantum gates.
Optical qudits can be realized using the various degrees of freedom of a single photon, such as polarization, momentum, frequency, timing or even the spatial shape of a singlephoton beam. In recent years it has become popular to use states that have definite values of orbital angular momentum (OAM) as qudits^{1}. However, OAM states are generally difficult to manipulate, as they require complex wavefront transformations usually implemented using holographic techniques. These techniques are rather slow, and thus limit the bit rates that can be achieved. This is a serious problem, particularly because efficient quantum communication requires high bit rates.
Fabio Sciarrino and coworkers^{2} now introduce a convenient method of transferring information from polarization to OAM and vice versa, thus providing a convenient and elegant way for fast polarization controllers to manipulate OAM qudits.
A natural basis for OAM states is a set of Laguerre–Gaussian modes, which are eigenmodes of the wave equation under the paraxial approximation. The wavefronts of such modes are composed of m intertwined helical surfaces with a phase singularity in the centre of the beam. Each photon in the beam has an OAM of value mħ. If the beam is circularly polarized, it also has a spin angular momentum of ±ħ per photon. Usually, when the beam propagates in a homogeneous and isotropic media, these two kinds of angular momentum are separately conserved. However, in a specially designed medium that is both inhomogeneous and spatially anisotropic, spin and orbital angular momentum may be transferred between each other, as shown by the authors several years ago^{3}. This occurs in particular if a circularly polarized beam passes through a slab of birefringent material with a fixed phase shift of π between the polarization components (as in a regular halfwave plate), but with the orientation of its fast axis (optical axis) varying according to the location of the beam on the plate. A 'qplate' is a plate in which the orientation of the optical axis obeys the relation α(r,φ) = qφ + φ_{0}, where r and φ are the radial and angular polar coordinates in the plane of the plate, and q and φ_{0} are constants. An example of a qplate with q = 1 is shown in Fig. 1. In this particular case, if a left (L)circularly polarized beam passes through the plate, it becomes right (R)circularly polarized and also acquires an OAM of 2ħ per photon. The authors made such a birefringent plate using a nematic liquidcrystal layer of an appropriate thickness placed between two polyamidecovered glass plates. Circular orientation of the liquid crystals was achieved by scratching the surface with a rotating piece of fabric.
When a qplate is used with single photons (that is, a nonclassical state of light), its unique property of transferring information from polarization to OAM (and vice versa) may be used to generate entanglement between these two degrees of freedom. In such an entangled state, the wavefunction describing the joint state cannot be factorized into separate spin and OAM components. Imagine a horizontally (H)polarized photon passing through a qplate. Because an Hpolarized state may be regarded as an equally weighted superposition of R and Lcircularly polarized states, the output of the system turns out to be a Bell state — a state exhibiting maximum entanglement^{4} — such that:
where the π and o subscripts denote polarization and OAM states, respectively.
Sciarrino et al. use this approach to generate fourdimensional qudits known as 'ququarts', which are encoded in both the OAM and polarization of a single photon. Among the various states in fourdimensional Hilbert space, a specific set of bases that are 'mutually unbiased' is of particular importance. These form a complete set of five orthogonal bases in such a way that a state ψ_{i}〉 belonging to any basis has equal projections onto the states of all the other bases φ_{j}〉, giving 〈ψ_{i}φ_{j}〉 = 1/4. Thus, all the other states look the same when observed from any basis — a property that is crucial for the security of quantum key distribution protocols. It can be shown that there are exactly five mutually unbiased bases in a fourdimensional Hilbert space, and that a quantum key distribution protocol becomes more secure as more of these bases are used. Until now, the problem has been that two of these bases consist of entangled states, which are difficult to manipulate — this is where the approach developed by Sciarrino et al. becomes particularly helpful. The authors have succeeded in preparing and analysing states belonging to all five mutually unbiased bases. At the preparation stage, a qubit encoded in photon polarization is transferred to OAM with a qplate, after which a second qubit is reencoded in polarization to obtain a hybrid ququart state. For detection, the polarization qubit is first read out with a standard polarization analyser, following which the OAM qubit is transferred to a polarization qubit and then analysed in the same way.
The scheme is attractive because it combines the rich capabilities of encoding information in OAM space with the ability to use simple and relatively fast polarization controllers as a manipulation tool. However, applications will probably be limited to freespace quantum communication as Laguerre–Gaussian modes are not the eigenmodes for ordinary optical fibres, making OAM encoding inapplicable for fibrebased communication.
References
 1
MolinaTerriza, G., Torres, J. P & Torner, L. Nature Phys. 3, 305–310 (2007).
 2
Nagali, E., Sansoni, L., Marrucci, L., Santamato, E. & Sciarrino, F. Phys. Rev. A 81, 052317 (2010).
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Marrucci, L., Manzo, C. & Paparo, D. Phys. Rev. Lett. 96, 163905 (2006).
 4
Nagali, E. et al. Phys. Rev. Lett. 103, 013601 (2009).
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Straupe, S., Kulik, S. The quest for higher dimensionality. Nature Photon 4, 585–586 (2010). https://doi.org/10.1038/nphoton.2010.215
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