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Light Science & Applications

Qudit W and Greenberger–Horne–Zeilinger states on a two-photon chip

Yulin Chi, Haiyu Ding, Fei Wang, Chengkang Pan, Chunju Shao

Published Aug 5, 2026• 0 citations
Publisher

Abstract

High-dimensional quantum entangled states exhibit unique properties. While generalizations like qudit Bell, Greenberger-Horne-Zeilinger (GHZ), and Cluster states have been studied, the qudit W states remain underexplored. We discover that a superposition of qudit Dicke states can serve as high-dimensional generalizations of W states, demonstrating their enhanced robustness against qudit loss, and present linearly scalable quantum circuits for their generation. Using a programmable silicon-photonic entanglement generator, we first theoretically and experimentally demonstrate the generation of 3-ququart GHZ and W states using two photons and one ancillary ququart, with fidelities of 0.932 ± 0.009 and 0.901 ± 0.012, respectively. Genuine multipartite entanglement of the GHZ state and the improved resilience of the W state are confirmed. Our findings provide new insights and efficient approaches for qudit-based quantum science and technologies.

DOI
10.1038/s41377-026-02285-7

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High-dimensional quantum entangled states exhibit unique properties. While generalizations like qudit Bell, Greenberger-Horne-Zeilinger (GHZ), and Cluster states have been studied, the qudit W states remain underexplored. We discover that a superposition of qudit Dicke states can serve as high-dimensional generalizations of W states, demonstrating their enhanced robustness against qudit loss, and present linearly scalable quantum circuits for their generation. Using a programmable silicon-photonic entanglement generator, we first theoretically and experimentally demonstrate the generation of 3-ququart GHZ and W states using two photons and one ancillary ququart, with fidelities of 0.932 ± 0.009 and 0.901 ± 0.012, respectively. Genuine multipartite entanglement of the GHZ state and the improved resilience of the W state are confirmed. Our findings provide new insights and efficient approaches for qudit-based quantum science and technologies.
Abstract Quantum technology, leveraging non-classical characteristics, enables breakthroughs in computation, communication, and metrology. Entanglement generation is fundamental. However, efficiently creating large-scale entanglement remains challenging. Using 2 n physical modes within a single particle has been exploited for encoding n -qubit entanglement, yet this approach is believed to be unscalable. Here, using sequentially quantum correlations on a single particle, we present a Ququart Encoding scheme to encode and process multi-qubit entanglement within ququart. We implement a programmable integrated-photonic entanglement generator on a silicon chip, encoding up to 5-qubit entanglement and quantum circuits within 2 ququarts. Within a ququart, we generate high-fidelity 3-qubit W and Greenberger-Horne-Zeilinger states and verify quantum teleportation and phase estimation. Encoding a pair of entangled photons as ququarts yields a 5-qubit Greenberger-Horne-Zeilinger state at 688 Hz, with fidelity 0.956 ± 0.053, confirming genuine 5-qubit entanglement. While the 2 n physical-mode overhead persists for general purposes, our scheme reduces it from exponential to linear under certain circumstances, potentially accelerating large-scale quantum systems.
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.
High-dimensional multipartite entanglement plays a crucial role in quantum information science. However, existing schemes for generating such entanglement become complex and costly as the dimension of quantum units increases. In this Letter, we overcome the limitation by proposing a significantly enhanced linear optical heralded scheme that generates the d-level N-partite Greenberger-Horne-Zeilinger (GHZ) state with single-photon sources and linear operations. Our scheme requires dN photons, which is the minimal required photon number, with substantially improved success probability from previous schemes. It employs linear optical logic gates compatible with any qudit encoding system and can generate generalized GHZ states with installments of beam splitters. With efficient generations of high-dimensional resource states, our work opens avenues for further exploration in high-dimensional quantum information processing.
High-dimensional qudit (i.e., ddd-level or ddd-state) systems outperform two-dimensional qubit (i.e., 2-level or 2-state) systems in some quantum information processing tasks. We exploit entanglement purification protocols (EPPs) for extracting a subset of high quality arbitrary ddd-dimensional nnn-partite Greenberger-Horne-Zeilinger (GHZ) states from a large set of less entangled GHZ states. In our protocols, qudit-flip and phase-flip errors can be corrected, and the fidelity of the output state can be asymptotically improved to unity by iterating the EPP process. Moreover, the schemes are immune to the number of polluted photons, the fidelity thresholds of the proposed EPPs are developed, and the spatial-based single-qudit operations can be well manipulated with a range of balanced beam splitters, and phase shifters. These features make the proposed schemes offer an alternative method for high-dimensional multipartite entanglement purification.
High-dimensional quantum systems, known as qudits with dimension d>2d>2d>2, possess the capability to encode and process more information than traditional two-dimensional qubits. Entangled states serve as crucial resources for both quantum computation and quantum communication. On the basis of circuit quantum electrodynamics (QED) systems, we introduce an efficient scheme to deterministically generate high-dimensional multipartite entangled states of hybrid qutrits (d=3d=3d=3). Assuming a simple initial state has already been prepared with the use of local operations, our method requires only a single step to produce hybrid Greenberger-Horne-Zeilinger (GHZ) entangled states among circuit QED systems consisting of superconducting qutrits and a superconducting cavity. Remarkably, the number of steps and the operational time remain constant as the number of qutrits increases, highlighting the efficiency and scalability of our approach. Using numerical simulations for a three-dimensional hybrid GHZ state of seven qutrits, we demonstrate the feasibility of our scheme and study the effects of various parameters on the fidelity. The importance of this research is in directing the experimental creation of multipartite, high-dimensional, and hybrid GHZ states, which can then be used to enhance quantum communication protocols and computational processes.
Multipartite entanglement is one of the core concepts in quantum information science with broad applications that span from condensed matter physics to quantum physics foundation tests. Although its most studied and tested forms encompass two-dimensional systems, current quantum platforms technically allow the manipulation of additional quantum levels. We report the experimental demonstration and certification of a high-dimensional multipartite entangled state in a superconducting quantum processor. We generate the three-qutrit Greenberger-Horne-Zeilinger state by designing the necessary pulses to perform high-dimensional quantum operations. We obtain the fidelity of $76\mathrm{%}\ifmmode\pm\else\textpm\fi{}1\mathrm{%}$, proving the generation of a genuine three-partite and three-dimensional entangled state. To this date, only photonic devices have been able to create and certify the entanglement of these high-dimensional states. Our work demonstrates that another platform, superconducting systems, is ready to exploit genuine high-dimensional entanglement and that a programmable quantum device accessed on the cloud can be used to design and execute experiments beyond binary quantum computation.