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Photonics Research

High-dimensional Bell-state preparation and characterization based on single-photon radial and azimuthal mode multiplexing

Feng-Pei Li, Feng-Pei Li, Shiyue Jia, Ting-Yu Yang, Jian-lan Chen, Han Zhang, Xi‐Lin Wang

Published Jun 22, 2026• 0 citations
Publisher

Abstract

Laguerre–Gaussian (LG) modes are characterized by two integer parameters, p and l , which correspond to the radial and azimuthal modes of photons. Both degrees of freedom are discrete and high-dimensional, and each can be used to construct a high-dimensional Hilbert space. Here, we use the radial-azimuthal modes of a single photon to construct two-dimensional and four-dimensional Bell states and experimentally realize full single-photon Bell-state characterization, thereby verifying the orthogonality and completeness of the Bell states. Subsequently, we measure the magnitude and phase of a four-dimensional superposition state in a 16-dimensional Hilbert space, obtaining results in good agreement with theoretical predictions, which further confirms the completeness of the four-dimensional Bell basis. Our experiment demonstrates that the radial and azimuthal modes of a single photon play equally important roles. Multiplexing these two degrees of freedom enables the construction of higher-dimensional Hilbert spaces and significantly enhances the information-carrying capacity of a single photon. Our results hold potential applications in the development of quantum communication protocols with high channel capacity.

DOI
10.1364/prj.599690

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Laguerre–Gaussian (LG) modes are characterized by two integer parameters, p and l , which correspond to the radial and azimuthal modes of photons. Both degrees of freedom are discrete and high-dimensional, and each can be used to construct a high-dimensional Hilbert space. Here, we use the radial-azimuthal modes of a single photon to construct two-dimensional and four-dimensional Bell states and experimentally realize full single-photon Bell-state characterization, thereby verifying the orthogonality and completeness of the Bell states. Subsequently, we measure the magnitude and phase of a four-dimensional superposition state in a 16-dimensional Hilbert space, obtaining results in good agreement with theoretical predictions, which further confirms the completeness of the four-dimensional Bell basis. Our experiment demonstrates that the radial and azimuthal modes of a single photon play equally important roles. Multiplexing these two degrees of freedom enables the construction of higher-dimensional Hilbert spaces and significantly enhances the information-carrying capacity of a single photon. Our results hold potential applications in the development of quantum communication protocols with high channel capacity.
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.
Spatially entangled twin photons allow the study of high-dimensional entanglement, and the Laguerre-Gauss modes are the most commonly used basis to discretize the single-photon mode spaces. In this basis, to date only the azimuthal degree of freedom has been investigated experimentally due to its fundamental and experimental simplicity. We show that the full spatial entanglement is indeed accessible experimentally; i.e., we have found practicable radial detection modes with negligible cross correlations. This allows us to demonstrate hybrid azimuthal-radial quantum correlations in a Hilbert space with more than 100 dimensions per photon.
Abstract It is well known that photons can carry a spatial structure akin to a ‘twisted’ or ‘rippled’ wavefront. Such structured light fields have sparked significant interest in both classical and quantum physics, with applications ranging from dense communications to light–matter interaction. Harnessing the full advantage of transverse spatial photonic encoding using the Laguerre–Gaussian (LG) basis in the quantum domain requires control over both the azimuthal (twisted) and radial (rippled) components of photons. However, precise measurement of the radial photonic degree-of-freedom has proven to be experimentally challenging primarily due to its transverse amplitude structure. Here we demonstrate the generation and certification of full-field LG entanglement between photons pairs generated by spontaneous parametric down conversion in the telecom regime. By precisely tuning the optical system parameters for state generation and collection, and adopting recently developed techniques for precise spatial mode measurement, we are able to certify fidelities up to 85% and entanglement dimensionalities up to 26 in a 43-dimensional radial and azimuthal LG mode space. Furthermore, we study two-photon quantum correlations between nine LG mode groups, demonstrating a correlation structure related to mode group order and inter-modal cross-talk. In addition, we show how the noise-robustness of high-dimensional entanglement certification can be significantly increased by using measurements in multiple LG mutually unbiased bases. Our work demonstrates the potential offered by the full spatial structure of the two-photon field for enhancing technologies for quantum information processing and communication.
We present a theoretical scheme for the complete Bell state measurement (BSM) for high-dimensional hyperentangled photons in polarization and path degrees of freedom (DOFs), in which the polarization entanglement is in two-dimensional Hilbert space and the path entanglement is in four-dimensional Hilbert space. This 2×4-dimensional photonic quantum system contains 64 mutually orthogonal high-dimensional hyperentangled Bell states, which can be unambiguously discriminated by using weak cross-Kerr nonlinearity, linear optical element, and single-photon detector. This is the first complete BSM scheme for high-dimensional hyperentangled photons in two DOFs assisted by the quantum nonlinear effect, and we also demonstrate its application in the quantum superdense coding protocol, which allows the transmission of log 2 64=6 bits of classical information via just sending one photon. Moreover, our scheme only requires the achievable small nonlinear phase shift, and it can provide new avenues for other BSM-based quantum information technologies involving photonic high-dimensional hyperentanglement.
The orbital angular momentum (OAM) carried by twisted photons provides a promising playground for high-dimensional quantum information processing. While Bell-state measurement is the cornerstone for various quantum information applications, the deterministic discrimination of the complete high-dimensional Bell states with linear optics remains relatively unexplored in the OAM state space. Here, we demonstrate a theoretical scheme for the complete four-dimensional OAM Bell-state measurement by using the single-photon hyperentangled state analyzer, in which the auxiliary two-dimensional polarization entanglement and two-dimensional path entanglement are utilized. Our scheme offers an alternative route toward enhancing the channel capacity in quantum communication and increasing the robustness against deleterious noise in practical experiments with twisted photons.
The Bell basis, a set of maximally entangled biphoton states, is a critical prerequisite towards quantum information processing, and many quantum applications have highlighted the requirement for manipulation of the high-dimensional Bell basis. While Bell states can be created by using ingenious single-photon quantum gates, their implementation complexity in higher dimensions is significantly increased. Here we present an elaborate approach to show that the adaptive pump modulation enables the efficient preparation of the Bell basis in arbitrary-dimensional Hilbert space. A complete set of four-dimensional orbital angular momentum Bell states is experimentally created, yielding high fidelities for certifying the entanglement dimensionality. Our strategy can be simply generalized to prepare more complex forms of quantum states, even exploiting other physical degrees of freedom. Also, it can facilitate the use of high-dimensional entanglement in a variety of quantum protocols, in particular those requiring quantum dense coding.