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EPJ Quantum Technology

Deterministic high-dimensional entangled states analysis and quantum teleportation

Fang‐Fang Du, Ling‐Hui Li, Chen Huang, Zhuo-Ya Bai, Qiulin Tan

Published May 22, 2026• 0 citations
Publisher

Abstract

Entanglement in high-dimensional systems plays a vital role in quantum information processing (QIP), as it extends the available Hilbert space, supports the manipulation of more intricate information, and broadens the scope of implementable quantum operations. In the paper, we present a deterministic scheme for the four-dimensional two-qudit Bell entangled state analysis for the four-photon system in the polarization degree of freedom, resorting to the weak cross-Kerr nonlinearity and some linear optical components. In our scheme, the analysis of four-dimensional Bell entangled state is constructed by three-photon Toffoli gate and quantum nondemolition detectors (QND 1 and QND 2 ). By phase shifts of the coherent beams, the 16 orthogonal four-dimensional two-qudit Bell states can be completely distinguished from each other. The proposal can be extended to the four-dimensional three-qudit entangled state analysis and n -qudit entangled state analysis. In addition, our deterministic scheme for high-dimensional entangled state analysis can be applied to high-dimensional quantum teleportation, including both uncontrolled and controlled quantum teleportation protocols. Moreover, our scheme can achieve high fidelity and efficiency, further broadening its applicability in high-dimensional QIP.

DOI
10.1140/epjqt/s40507-026-00525-8

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In this paper, we present an efficient scheme for the complete analysis of the four-dimensional Bell state of the photon system in the path degree of freedom, resorting to the weak cross-Kerr nonlinearity and linear optics. In our scheme, the measurements on two probe coherent states are used for obtaining the bit information of high-dimensional entanglement, and the linear optical elements and single-photon detectors are used for obtaining the relative phase information of high-dimensional entanglement. With these two independent processes, the 16 orthogonal four-dimensional path Bell states can be completely distinguished from each other. This simple and realizable scheme may have useful applications in the quantum information technology based on high-dimensional entanglement.
The generation of a high-dimensional quantum entangled state provides a crucial resource for achieving more complex and diverse quantum information processing tasks. In this paper, we present two deterministic protocols designed for the generation of a four-dimensional two-qudit Bell state and three-qudit Greenberger-Horne-Zeilinger (GHZ) state assisted by cross-Kerr nonlinearity, where the four-dimensional qudit is encoded in the hybrid polarization-spatial state of the single photon to conserve quantum resources and alleviate environmental noise. The proposed protocols for generating a four-dimensional two-qudit Bell state and three-qudit GHZ state are circuit concise, high success, robust fidelity, noise resistant, and resource saving. Moreover, the four-dimensional NNN-qudit (N>3N>3N>3) GHZ states can be generated with the help of the $N\ensuremath{-}1$ generating devices of the four-dimensional Bell state. Consequently, its efficiency and fidelity decay exponentially by a factor of $N\ensuremath{-}1$. Further, the arbitrary high-dimensional qudit-encoded entangled states can be achieved with linear-optics four-dimensional single-qudit XXX gates and ZZZ gates. These protocols offer a reference for further advancements in high-dimensional state generation technology and providing insights for the practical implementations of metrological enhancement, quantum simulation, high-dimensional quantum computation, and quantum error correction.
We present a deterministic nondestructive hyperentangled Bell state analysis protocol for photons entangled in three degrees of freedom (DOFs), including polarization, spatial-mode, and time-bin DOFs. The polarization Bell state analyzer and spatial-mode Bell state analyzer are constructed by polarization parity-check quantum nondemolition detector (P-QND) and spatial-mode parity-check quantum nondemolition detector (S-QND) using cross-Kerr nonlinearity, respectively. The time-bin Bell state analyzer is constructed by the swap gate for polarization state and time-bin state of a photon (P-T swap gate) and P-QND. The Bell states analyzer for one DOF will not destruct the Bell states of other two DOFs, so the polarization-spatial-time-bin hyperentangled Bell states can be determinately distinguished without destruction. This deterministic nondestructive state analysis method has useful applications in quantum information protocols.
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.
Hyperentangled state analysis is an important module in high-capacity quantum communication. We present a self-assisted deterministic hyperentangled-Bell-state analysis (HBSA) scheme for photon system entangled in three degrees of freedom (DOFs), where 64 polarization-double longitudinal momentum hyperentangled Bell states are completely distinguished. In this HBSA scheme, the four first longitudinal momentum Bell states are distinguished determinately by nondestructive first longitudinal momentum Bell state analyzer, which is constructed with cross-Kerr nonlinearity medium. The 16 second longitudinal momentum-polarization hyperentangled Bell states are distinguished determinately by self-assisted second longitudinal momentum-polarization hyperentangled Bell state analyzer using linear optical elements, where the first longitudinal momentum Bell state and time-bin entangled state are used as auxiliary. Using this self-assisted method, the application of nonlinear optical resource in HBSA scheme has been largely reduced, which makes this self-assisted deterministic HBSA scheme has potential application prospects in high-capacity quantum communication.
Abstract Remote implementation of high-dimensional quantum operation is pivotal for advancing quantum communication and distributed computing. Here, we propose a deterministic protocol capable of performing arbitrary four-dimensional unitary operations between two spatially separated nodes. The scheme utilizes hybrid encoding in both polarization and spatial degrees of freedom, which not only conserves quantum resources but also leverages the scalability of high-dimensional photonic systems. A key feature of the protocol is the use of weak cross-Kerr nonlinearities combined with X -quadrature homodyne detection. This design significantly reduces the reliance on strong nonlinear interactions and enhances robustness against photon loss. By performing classical feed-forward operations conditioned on coherent-state measurement outcomes, our approach ensures deterministic execution and high operational fidelity. Importantly, the protocol avoids the need for complex ancillary entanglement by relying solely on linear optics and simplified measurements. Due to the use of mature optical components and a loss-resilient architecture, numerical simulations under realistic experimental conditions indicate a high success probability (≈1) of this scheme. These advantages highlight the practicality and resource efficiency of our method, making it well-suited for the implementation of high-dimensional quantum communication protocols with current photonic technologies.