High-dimensional encoding of quantum information provides a promising method of transcending current limitations in quantum communication. One of the central challenges in the pursuit of such an approach is the certification of high-dimensional entanglement. In particular, it is desirable to do so without resorting to inefficient full state tomography. Here, we show how carefully constructed measurements in two bases (one of which is not orthonormal) can be used to faithfully and efficiently certify bipartite high-dimensional states and their entanglement for any physical platform. To showcase the practicality of this approach under realistic conditions, we put it to the test for photons entangled in their orbital angular momentum. In our experimental setup, we are able to verify 9dimensional entanglement for a pair of photons on a 11-dimensional subspace each, at present the highest amount certified without any assumptions on the state.Quantum communication offers advantages such as enhanced security in quantum key distribution (QKD) protocols [1] and increased channel capacities [2] with respect to classical means of communication. All of these improvements, ranging from early proposals [3] to recent exciting developments such as fully device-independent QKD [4, 5], rely on one fundamental phenomenon: quantum entanglement. Currently, the workhorse of most implementations is entanglement between qubits, i.e., between two-dimensional quantum systems (e.g., photon polarization). However, it has long been known that higher-dimensional entanglement can be useful in overcoming the limitations of qubit entanglement [6, 7], offering better key rates [8], higher noise resistance [9, 10] and improved security against different attacks [11].Attempting to capitalize on this insight, recent experiments have successfully generated and certified highdimensional entanglement in different degrees of freedom. In particular, the canonical way of generating two-dimensional polarization entanglement in downconversion processes already offers the potential for exploring entanglement in higher dimensions. This can be achieved by exploiting spatial degrees of freedom [12, 13], orbital angular momentum (OAM) [14-16], energy-time based encodings [17][18][19][20], or combinations thereof in hyper-entangled quantum systems [21,22]. High-dimensional quantum systems have recently also been explored in matter-based systems such as Cesium atoms [23] and superconducting circuits [24]. Thus, high- dimensional quantum systems are not only of fundamental interest but are also becoming more readily available.In this context, the certification and quantification of entanglement in many dimensions is a crucial challenge since full state tomography (FST) for bipartite systems of local dimension d requires measurements in (d + 1) 2 global product bases (i.e., tensor product bases for the global state) [25], which quickly becomes impractical in high dimensions. Due to the complexity of realizing measurements in high-dimensional spaces, previous ex...
With the emergence of the field of quantum communications, the appropriate choice of photonic degrees of freedom used for encoding information is of paramount importance. Highly precise techniques for measuring the polarisation, frequency, and arrival time of a photon have been developed. However, the transverse spatial degree of freedom still lacks a measurement scheme that allows the reconstruction of its full transverse structure with a simple implementation and a high level of accuracy. Here we show a method to measure the azimuthal and radial modes of Laguerre-Gaussian beams with a greater than 99 % accuracy, using a single phase screen. We compare our technique with previous commonly used methods and demonstrate the significant improvements it presents for quantum key distribution and state tomography of high-dimensional quantum states of light. Moreover, our technique can be readily extended to any arbitrary family of spatial modes, such as mutually unbiased bases, Hermite-Gauss, and Ince-Gauss. Our scheme will significantly enhance existing quantum and classical communication protocols that use the spatial structure of light, as well as enable fundamental experiments on spatial-mode entanglement to reach their full potential.
The transfer of quantum information through a noisy environment is a central challenge in the fields of quantum communication, imaging, and nanophotonics. In particular, high-dimensional quantum states of light enable quantum networks with significantly higher information capacities and noise-robustness as compared with qubits. However, while qubit-entanglement has been distributed over large distances through free-space and fibre, the transport of high-dimensional entanglement is hindered by the complexity of the channel, which encompasses effects such as free-space turbulence or mode-mixing in multi-mode waveguides. Here we demonstrate the transport of six-dimensional spatial-mode entanglement through a two-metre long, commercial multi-mode fibre with 84.4% fidelity. We show how the entanglement can itself be used to measure the transmission matrix of the complex medium, allowing the recovery of quantum correlations that were initially lost. Using a unique property of entangled states, the medium is rendered transparent to entanglement by carefully "scrambling" the photon that did not enter it, rather than unscrambling the photon that did. Our work overcomes a primary challenge in the fields of quantum communication and imaging, and opens a new pathway towards the control of complex scattering processes in the quantum regime.
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