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Coherence and correlations represent two related properties of a compound system. The system can be, for instance, the polarization of a photon, which forms part of a polarization-entangled two-photon state, or the spatial shape of a coherent beam, where each spatial mode bears different polarizations. Whereas a local unitary transformation of the system does not affect its coherence, global unitary transformations modifying both the system and its surroundings can enhance its coherence, transforming mutual correlations into coherence. The question naturally arises of what is the best measure that quantifies the correlations that can be turned into coherence, and how much coherence can be extracted. We answer both questions, and illustrate its application for some typical simple systems, with the aim at illuminating the general concept of enhancing coherence by modifying correlations. Introduction.-Coherence is one of the most important concepts needed to describe the characteristics of a stream of photons [1, 2], where it allows us to characterize the interference capability of interacting fields. However its use is far more general as it plays a striking role in a whole range of physical, chemical, and biological phenomena [3]. Measures of coherence can be implemented using classical and quantum ideas, which lead to the question of in which sense quantum coherence might deviate from classical coherence phenomena [4], and to the evaluation of measures of coherence [5][6][7].Commonly used coherence measures consider a physical system as a whole, omitting its structure. The knowledge of the internal distribution of coherence between subsystems and their correlations becomes necessary for predicting the evolution (migration) of coherence in the studied system. The evolution of a twin beam from the near field into the far field represents a typical example occurring in nature [8]. The creation of entangled states by merging the initially separable incoherent and coherent states serves as another example [7]. Or, in quantum computing the controlled-NOT gate entangles (disentangles) two-qubit states [9,10], at the expense (in favor) of coherence. Many quantum metrology and communication applications benefit from correlations of entangled photon pairs originating in spontaneous parametric down-conversion [11][12][13]. Even separable states of photon pairs, i.e. states with suppressed correlations, are very useful, e.g., in the heralded single photon sources [14,15]. For all of these, and many others, examples the understanding of common evolution of coherence and correlations is crucial.The Clauser-Horne-Shimony-Holt (CHSH) Bell's-like inequality [16][17][18] has been usually considered to quantify nonclassical correlations present between physically sepa-

We demonstrate a different scheme to perform optical sectioning of a sample based on the concept of induced coherence [Zou et al., Phys. Rev. Lett. 67, 318 (1991)]. This can be viewed as a different type of optical coherence tomography scheme where the varying reflectivity of the sample along the direction of propagation of an optical beam translates into changes of the degree of first-order coherence between two beams. As a practical advantage the scheme allows probing the sample with one wavelength and measuring photons with another wavelength. In a bio-imaging scenario, this would result in a deeper penetration into the sample because of probing with longer wavelengths, while still using the optimum wavelength for detection. The scheme proposed here could achieve submicron axial resolution by making use of nonlinear parametric sources with broad spectral bandwidth emission.

The global quantum network requires the distribution of entangled states over long distances, with significant advances already demonstrated using entangled polarisation states, reaching approximately 1200 km in free space and 100 km in optical fibre. Packing more information into each photon requires Hilbert spaces with higher dimensionality, for example, that of spatial modes of light. However spatial mode entanglement transport requires custom multimode fibre and is limited by decoherence induced mode coupling. Here we transport multi-dimensional entangled states down conventional single-mode fibre (SMF). We achieve this by entangling the spin-orbit degrees of freedom of a bi-photon pair, passing the polarisation (spin) photon down the SMF while accessing multi-dimensional orbital angular momentum (orbital) subspaces with the other. We show high fidelity hybrid entanglement preservation down 250 m of SMF across multiple 2 × 2 dimensions, demonstrating quantum key distribution protocols, quantum state tomographies and quantum erasers. This work offers an alternative approach to spatial mode entanglement transport that facilitates deployment in legacy networks across conventional fibre.

Abstract:We demonstrate experimentally that spontaneous parametric down-conversion in an Al x Ga 1−x As semiconductor Bragg reflection waveguide can make for paired photons highly entangled in the polarization degree of freedom at the telecommunication wavelength of 1550 nm. The pairs of photons show visibility higher than 90% in several polarization bases and violate a Clauser-Horne-Shimony-Holt Bell-like inequality by more than 3 standard deviations. This represents a significant step toward the realization of efficient and versatile self pumped sources of entangled photon pairs on-chip.

One of the most captivating properties of diffraction-free optical fields is their ability to reconstruct upon propagation in the presence of an obstacle both, classically and in the quantum regime. Here we demonstrate that the local entanglement, or non-separability, between the spatial and polarisation degrees of freedom also experience self-healing. We measured and quantified the degree of nonseparability between the two degrees of freedom when propagating behind various obstructions, which were generated digitally. Experimental results show that even though the degree of nonseparability reduces after the obstruction, it recovers to its maximum value within the classical selfhealing distance. To confirm our findings, we performed a Clauser-Horne-Shimony-Holt Bell-like inequality measurement, proving the self-reconstruction of non-separability. These results indicate that local entanglement between internal degrees of freedom of a photon, can be recovered by suitable choice of the enveloping wave function. I. INTRODUCTIONSelf-healing is one of the most fascinating properties of diffraction-free optical fields [1]. These fields have the ability to reconstruct if they are partially disturbed by an obstruction placed in their propagation path. Diffraction-free beams have found applications in fields such as imaging [2][3][4], optical trapping [5][6][7][8], laser material processing [9], amongst many others. Arguably, the most well-known propagation invariant (self-healing) fields are Bessel modes of light, first introduced in 1987 by J. Durin [1,10]. However, the self-healing property is not limited to so called non-diffracting beams, but also appears in helico-conical [11], caustic, or self-similar fields, namely Airy [12], Pearcey [13], Laguerre-Gaussian [14,15] and even standard Gaussian beams [16]. Furthermore, within the last years, it has been shown that self-healing can also be observed at the quantum level, for example, McLaren et al. demonstrated experimentally the self-reconstruction of quantum entanglement [17]. Importantly, self-healing is not only an attribute of scalar fields but it can also apply to beams with spatially variant polarization [18][19][20]. Bessel beams also appear as complex vector light fields, where polarisation and spatial shape can be coupled in a non-separable way [21][22][23]. This property has fueled a wide variety of applications, from industrial processes, such as, drilling or cutting [9,24,25], to optical trapping [26][27][28][29][30][31], high resolution microscopy [32], quantum and classical communication [33][34][35], amongst many others. Controversially, such non-separable states of classical light are sometimes referred to as classically or nonquantum entangled [36]. This stems from the fact that the quintessential property of quantum entanglement is non-separability, which is not limited to quantum systems. Indeed, the equivalence has been shown to be more than just a mathematical construct [34]. While such classical non-separable fields do not exhibit non-locality, they

We demonstrate a scheme to generate noncoherent and coherent correlations, i.e., a tunable degree of entanglement, between degrees of freedom of a single photon. Its nature is analogous to the tuning of the purity (first-order coherence) of a single photon forming part of a two-photon state by tailoring the correlations between the paired photons. Therefore, well-known tools such as the Clauser-Horne-Shimony-Holt (CHSH) Bell-like inequality can also be used to characterize entanglement between degrees of freedom. More specifically, CHSH inequality tests are performed, making use of the polarization and the spatial shape of a single photon. The four modes required are two polarization modes and two spatial modes with different orbital angular momentum.