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Physical Review Letters

Two-Measurement Tomography of High-Dimensional Orbital Angular Momentum Entanglement

Yi Li, Shuang‐Yin Huang, Min Wang, Chenghou Tu, Xi‐Lin Wang, Yongnan Li, Hui‐Tian Wang

Published Feb 3, 2023• 32 citations
Publisher

Abstract

High-dimensional (HD) entanglement enables an encoding of more bits than in the two-dimensional case and promises to increase communication capacity over quantum channels and to improve robustness to noise. In practice, however, one of the central challenges is to devise efficient methods to quantify the HD entanglement explicitly. Full quantum state tomography is a standard technology to obtain all the information about the quantum state, but it becomes impractical because the required measurements increase exponentially with the dimension in HD systems. Hence, it is highly anticipated that a new method will be found for characterizing the HD entanglement with as few measurements as possible and without introducing unwarranted assumptions. Here, we present and demonstrate a scan-free tomography method independent of dimension, which only requires two measurements for the characterization of two-photon HD orbital angular momentum (OAM) entanglement. Taking Laguerre-Gaussian modes of photons as an example, the density matrices of OAM entangled states are experimentally reconstructed with very high fidelity. Our method is also generalized to the mixed HD OAM entanglement. Our results provide realistic approaches for quantifying more complex OAM entanglement in many scientific and engineering fields such as multiphoton HD quantum systems and quantum process tomography.

DOI
10.1103/physrevlett.130.050805

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High-dimensional (HD) entanglement enables an encoding of more bits than in the two-dimensional case and promises to increase communication capacity over quantum channels and to improve robustness to noise. In practice, however, one of the central challenges is to devise efficient methods to quantify the HD entanglement explicitly. Full quantum state tomography is a standard technology to obtain all the information about the quantum state, but it becomes impractical because the required measurements increase exponentially with the dimension in HD systems. Hence, it is highly anticipated that a new method will be found for characterizing the HD entanglement with as few measurements as possible and without introducing unwarranted assumptions. Here, we present and demonstrate a scan-free tomography method independent of dimension, which only requires two measurements for the characterization of two-photon HD orbital angular momentum (OAM) entanglement. Taking Laguerre-Gaussian modes of photons as an example, the density matrices of OAM entangled states are experimentally reconstructed with very high fidelity. Our method is also generalized to the mixed HD OAM entanglement. Our results provide realistic approaches for quantifying more complex OAM entanglement in many scientific and engineering fields such as multiphoton HD quantum systems and quantum process tomography.
Abstract High‐dimensional (HD) entanglement of photonic orbital angular momentum (OAM) offers significant potential for enhancing channel capacity and improving noise resistance in quantum information processing. However, the challenge of achieving simple and rapid measurement has limited its practical applications. In this work, a quantum state tomography (QST) framework is demonstrated that utilizes convolutional neural networks to rapidly reconstruct the density matrix of OAM entanglement from only two coincidence measurements. The experimental results for a 5D OAM entangled state yield a fidelity of 0.973 ± 0.005. This method is also applicable to mixed OAM entangled states and scenarios with incomplete tomographic measurements. These findings represent a significant step toward implementing high‐speed QST for applications involving HD spatial mode quantum state, whether in free space or integrated systems.
High-dimensional (HD) entanglement provides a very promising way of transcending the limitations of the two-dimensional entanglement between qubits for increasing channel capacity in many quantum protocols. In the pursuit of capitalizing on the HD entangled states, one of the central issues is to unambiguously and comprehensively quantify and reconstruct them. The full quantum state tomography is a unique solution, but it is undesirable and even impractical because the measurements increase rapidly in d 4 for a bipartite d-dimensional quantum state. Here we present a very efficient and practical tomography method—asymptotical locking tomography (ALT), which can harvest full information of bipartite d-dimensional entangled states by very few measurements less than 2d 2 only. To showcase the validity and reasonableness of our ALT, we carry out the test with the two-photon spin-orbital angular momentum hyperentangled states in a four-dimensional subspace. Besides high-efficiency and practicality, our ALT is also universal and can be generalized into multipartite HD entanglement and other quantum systems.
ENGLISH ABSTRACT: High-dimensional quantum entanglement offers an increase in information capacity per photon; a highly desirable property for quantum information processes such as quantum communication, computation and teleportation. As the orbital angular momentum (OAM) modes of light span an infinite-dimensional Hilbert space, they have become frontrunners in achieving entanglement in higher dimensions. In light of this, we investigate the potential of OAM entanglement of photons by controlling the parameters in both the generation and measurement systems. We show the experimental procedures and apparatus involved in generating and measuring entangled photons in two-dimensions. We verify important quantum tests such as the Einstein, Podolsky and Rosen (EPR) paradox using OAM and angle correlations, as well as a violation of a Bell-type inequality. By performing a full state tomography, we characterise our quantum state and show we have a pure, highly entangled quantum state. We demonstrate that this method can be extended to higher dimensions. The experimental techniques used to generate and measure OAM entanglement place an upper bound on the number of accessible OAM modes. As such, we investigate new methods in which to increase the spiral bandwidth of our generated quantum state. We alter the shape of the pump beam in spontaneous parametric down-conversion and demonstrate an effect on both OAM and angle correlations. We also made changes to the measurement scheme by projecting the photon pairs into the Bessel-Gaussian (BG) basis and demonstrate entanglement in this basis. We show that this method allows the measured spiral bandwidth to be optimised by simply varying the continuous radial parameter of the BG modes. We demonstrate that BG modes can be entangled in higher dimensions compared with the commonly used helical modes by calculating and comparing the linear entropy and fidelity for both modes. We also show that quantum entanglement can be accurately simulated using classical light using back-projection, which allows the study of projective measurements and predicts the strength of the coincidence correlations in an entanglement experiment. Finally, we make use of each of the techniques to demonstrate the effect of a perturbation on OAM entanglement measured in the BG basis. We investigate the self-healing property of BG beams and show that the classical property is translated to the quantum regime. By calculating the concurrence, we see that measured entanglement recovers after encountering an obstruction.
Abstract High-dimensional (HD) entanglement of photonic orbital angular momentum (OAM) is pivotal for advancing quantum communication and information processing, but its characterization remains significant challenges due to the complexity of quantum state tomography and experimental limitations such as low photon counts caused by losses. Here, we propose a pre-trained physics-informed neural network (PTPINN) framework that enables efficient and rapid reconstruction of HD-OAM entangled states under low photon counts. Experimental results show that the fidelity of five-dimensional OAM entanglement reaches F = 0.958 ± 0.010 even with an exposure time as short as 50 ms. This highlights the capability of PTPINN to achieve high-precision quantum state reconstruction with limited photons, owing to its innovative designs, thus overcoming the reliance on high photon counts typical of traditional methods. Our method provides a practical and scalable solution for high-fidelity characterization of HD-OAM entanglement in environments with low photon numbers and high noise, paving the way for robust long-distance quantum information transmission.
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 9-dimensional entanglement for a pair of photons on a 11-dimensional subspace each, at present the highest amount certified without any assumptions on the state.
Photons with a spiral phase will carry orbital angular momentum (OAM), which can serve as a valuable resource for constructing high-dimensional Hilbert spaces due to its orthogonality and unbounded dimensionality. However, the fast measurement of high-dimensional OAM spectra remains a challenge. While intensity detection is relatively straightforward, full characterization requires phase retrieval. Although quantum state tomography allows full reconstruction, it demands a number of measurements scaling quadratically with the d2 of dimension (d), leading to a “curse of dimensionality” as the dimension d increases. Thus, there is an urgent need for efficient and simple methods for extracting phase information from high-dimensional OAM spectra. Here, we propose a generalized Poincaré sphere model with analog Stokes-like parameters, enabling full phase measurement of a d-dimensional OAM spectrum using only 4d measurements, significantly reducing the data acquisition requirement. We experimentally validated the method by performing intensity and phase measurements on 8-, 16-, and 32-dimensional OAM comb spectra. The results show excellent agreement with theoretical predictions, with measured fidelities ranging from 0.9829 to 0.9965. This approach facilitates an efficient characterization of high-dimensional OAM spectra and contributes to advancing their practical applications.