We present an experimental study of higher-dimensional quantum key distribution protocols based on mutually unbiased bases, implemented by means of photons carrying orbital angular momentum. We perform (d + 1) mutually unbiased measurements in a classical prepare and measure scheme and on a pair of entangled photons for dimensions ranging from d = 2 to 5. In our analysis, we pay attention to the detection efficiency and photon pair creation probability. As security measures, we determine from experimental data the average error rate, the mutual information shared between the sender and receiver and the secret key generation rate per photon. We demonstrate that increasing the dimension leads to an increased information capacity as well as higher key generation rates per photon up to a dimension of d = 4.

Abstract:That the speed of light in free space is constant is a cornerstone of modern physics. However, light beams have finite transverse size, which leads to a modification of their wavevectors resulting in a change to their phase and group velocities. We study the group velocity of single photons by measuring a change in their arrival time that results from changing the beam's transverse spatial structure. Using time-correlated photon pairs we show a reduction of the group velocity of photons in both a Bessel beam and photons in a focused Gaussian beam. In both cases, the delay is several micrometers over a propagation distance of the order of 1 m. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves. Main textThe speed of light is trivially given as / , where is the speed of light in free space and is the refractive index of the medium. In free space, where = 1, the speed of light is simply . We show that the introduction of transverse structure to the light beam reduces the group velocity by an amount depending upon the aperture of the optical system. The delay corresponding to this reduction in the group velocity can be greater than the optical wavelength and consequently should not be confused with the HÀ Gouy phase shift (1, 2). To emphasize that this effect is both a linear and intrinsic property of light, we measure the delay as a function of the transverse spatial structure of single photons.The slowing down of light that we observe in free space should also not be confused with slow, or indeed fast, light associated with propagation in highly nonlinear or structured materials (3,4). Even in the absence of a medium, the modification of the speed of light has previously been known. For example, within a hollow waveguide, the wavevector along the guide is reduced below the free-space value, leading to a phase velocity greater than . Within the hollow waveguide, the product of the phase and group velocities is given as , = 2 , thereby resulting in a group velocity , along the waveguide less than (5). 2Although this relation for group and phase velocities is derived for the case of a hollow waveguide, the waveguide material properties are irrelevant. It is the transverse spatial confinement of the field that leads to a modification of the axial component of the wavevector, . In general, for light of wavelength , the magnitude of the wavevector, 0 = 2 / , and its Cartesian components { , , } are related through (5)All optical modes of finite , spatial extent require non-zero and , which implies < 0 , giving a corresponding modification of both the phase and group velocities of the light. In this sense, light beams with non-zero k x and k y are naturally dispersive, even in free space. Extending upon the case of a mode within a hollow waveguide, an example of a structured beam is a Bessel beam (Fig. 1A), which is itself the description of a mode within a circular waveguide (1, 6). In free space, Bessel beams can be created using an axicon, or its dif...

Any practical experiment utilising the innate D-dimensional entanglement of the orbital angular momentum (OAM) state space of photons is subject to the modal capacity of the detection system. We show that given such a constraint, the number of measured, entangled OAM modes in photon pairs generated by spontaneous parametric down-conversion (SPDC) can be maximised by tuning the phase-matching conditions in the SPDC process. We demonstrate a factor of 2 increase on the half-width of the OAM-correlation spectrum, from 10 to 20, the latter implying ≈ 50 -dimensional two-photon OAM entanglement. Exploiting correlations in the conjugate variable, angular position, we measure concurrence values 0.96 and 0.90 for two phase-matching conditions, indicating bipartite, D-dimensional entanglement where D is tuneable.PACS numbers: 03.65. Ud, 03.67.Bg, 03.67.Mn Much attention has been directed to the twodimensional state space of photon polarisation which provides both a conceptually and experimentally accessible playground [1][2][3]. D-dimensional two-photon entanglement, wherein each photon is a D-level quDit taking on any of D possible values, is an even more fertile playground. From a fundamental standpoint, higherdimensional entanglement implies stronger violations of locality [4,5] and is especially useful in the study of mutually unbiased bases in higher dimensions [6]. More relevant to practical applications, higher-dimensional entanglement provides higher information capacity [7,8] and increased security and robustness [8,9]. Experimentally, D-levels in photons can be achieved by using the temporal and spectral degrees of freedom [10], polarisation of more than one photon [11], transverse spatial profile [7], position and linear momentum [12], and angular position and orbital angular momentum [13].The entanglement of orbital angular momentum (OAM) in photons generated via spontaneous parametric down-conversion (SPDC) is firmly established theoretically and experimentally [14,15]. The interest in OAM stems from its discrete and theoretically infinitedimensional Hilbert space. Since the pioneering experiment of Zeilinger and co-workers ten years ago, OAM and it conjugate variable, angular position, has been steadily gaining ground as a mainstream variable in which to observe quantum correlations. Bell-type and Leggett inequalities have both been violated in twodimensional OAM subspaces analogous to the experiments done previously for polarisation [16,17]. The innate high-dimensional nature of OAM entanglement has been verified in an Einstein-Podolsky-Rosen (EPR) type experiment which measured both OAM and angular position [13]. A Bell-type inequality for higher dimensions has been recently violated using OAM states demonstrating experimental, two-photon, 11-dimensional entanglement [5]. The number of entangled OAM states that can be measured, i.e. the measurement spiral bandwidth depends on both the detection capability and the number of OAM states that is generated by the down-conversion process, i.e. the generation...

Mutually unbiased bases (MUBs) play a key role in many protocols in quantum science, such as quantum key distribution. However, defining MUBs for arbitrary high-dimensional systems is theoretically difficult, and measurements in such bases can be hard to implement. We show experimentally that efficient quantum state reconstruction of a high-dimensional multipartite quantum system can be performed by considering only the MUBs of the individual parts. The state spaces of the individual subsystems are always smaller than the state space of the composite system. Thus, the benefit of this method is that MUBs need to be defined for the small Hilbert spaces of the subsystems rather than for the large space of the overall system. This becomes especially relevant where the definition or measurement of MUBs for the overall system is challenging. We illustrate this approach by implementing measurements for a high-dimensional system consisting of two photons entangled in the orbital angular momentum degree of freedom, and we reconstruct the state of this system for dimensions of the individual photons from d = 2 to 5.

In quantum information, quantum systems and their properties offer unprecedented opportunities. Being able to harness additional degrees of freedom adds power and flexibility to quantum algorithms and protocols. In this work, we demonstrate that the radial transverse mode of a single photon constitutes one such degree of freedom. We do so by showing that we can tune the two-photon interference, a quintessential quantum effect and the basic constituent of many quantum protocols, by manipulating its radial transverse modal profiles. Our work, in addition to allowing for greater versatility of existing protocols and significantly increasing the information channel capacity, can inspire novel quantum information tasks.

Optimal quantum cloning is the process of making one or more copies of an arbitrary unknown input quantum state with the highest possible fidelity. All reported demonstrations of quantum cloning have so far been limited to copying two-dimensional quantum states, or qubits. We report the experimental realization of the optimal quantum cloning of four-dimensional quantum states, or ququarts, encoded in the polarization and orbital angular momentum degrees of freedom of photons. Our procedure, based on the symmetrization method, is also shown to be generally applicable to quantum states of arbitrarily high dimension-or qudits-and to be scalable to an arbitrary number of copies, in all cases remaining optimal. Furthermore, we report the bosonic coalescence of two single-particle entangled states.

Quantum entanglements between integer-order and fractional-order orbital angular momentums (OAMs) have been previously discussed. However, the entangled nature of arbitrary rational-order OAM has long been considered a myth due to the absence of an effective strategy for generating arbitrary rational-order OAM beams. Therefore, we report a single metadevice comprising a bilaterally symmetric grating with an aperture, creating optical beams with dynamically controllable OAM values that are continuously varying over a rational range. Due to its encoded spiniform phase, this novel metagrating enables the production of an average OAM that can be increased without a theoretical limit by embracing distributed singularities, which differs significantly from the classic method of stacking phase singularities using fork gratings. This new method makes it possible to probe the unexplored niche of quantum entanglement between arbitrarily defined OAMs in light, which could lead to the complex manipulation of microparticles, high-dimensional quantum entanglement and optical communication. We show that quantum coincidence based on rational-order OAM-superposition states could give rise to low cross-talks between two different states that have no significant overlap in their spiral spectra. Additionally, future applications in quantum communication and optical micromanipulation may be found.

Recent results in deeply subwavelength thickness films demonstrated coherent control and logical gate operations with both classical and single photon light sources. However, quantum processing and devices typically involve more than one photon and non-trivial input quantum states. Here we experimentally investigate two-photon N00N state coherent absorption in a multilayer graphene film. Depending on the N00N state input phase, it is possible to selectively choose between single or two photon absorption of the input state in the graphene film. These results demonstrate that coherent absorption in the quantum regime exhibits unique features opening up applications in multiphoton spectroscopy and imaging. Introduction: Coherent absorption is the process by which a partially absorbing material placed in a standing wave pattern can either perfectly transmit or absorb all of the incoming light. This process occurs as a result of the coherent interaction and of the relative phase relation of the two counter-propagating waves. Originally demonstrated in thick slabs of material where the process was likened to a "time-reversed laser" [1, 2]. Coherent absorption was recently extended to deeply subwavelength thickness (2D) materials that have 50% absorption [3,4,6]. Depending on the relative phase of the incident beams, it is possible to totally absorb (or transmit) light, thus providing a method to go beyond the theoretical limit of 50% absorption in a thin film [5], and therefore provide a route to obtain optical gates that rely on absorption [7]. This process relies heavily on the relations between reflection and transmission coefficients, r and t, and absorption rate, α, of the sample [8]. A special case is found for a sub-wavelength material exhibiting precisely 50% absorption, where complete absorption (or transmission) of the energy of the two beams is possible as a result of the relation between reflection and transmission coefficients being forced to r = ±t [5]. This has been shown experimentally with classical continuous wave sources [6], ultrashort pulses [9], and has also been extended to the nonlinear regime to achieve coherent control of nonlinear wave-mixing processes [10]. However, only a few studies have looked at the quantum nature of coherent absorption. Recent experimental work has shown that a single photon also exhibits coherent perfect absorption [11,12], thus implying for example, deterministic absorption of the single photon itself. More intriguing results are expected when non-trivial quantum states interact with a lossy beamsplitter. An N=2 N00N state, (|N, 0 + e iN φ |0, N )/ √ 2 (obtained as