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arXiv:2407.19735 · Quantum Science and Technology

Scalable high-dimensional multipartite entanglement with trapped ions

Harsh Vardhan Upadhyay, Sanket Tripathy, Ting Rei Tan, Baladitya Suri, Athreya Shankar

Published Jan 20, 2025• 0 citations
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Abstract

Abstract We propose a protocol for the preparation of generalized Greenberger–Horne–Zeilinger (GHZ) states of N atoms each with d = 3 or 4 internal levels. We generalize the celebrated one-axis twisting (OAT) Hamiltonian for N qubits to qudits by including OAT interactions of equal strengths between every pair of qudit levels, a protocol we call as balanced OAT (BOAT). Analogous to OAT for qubits, we find that starting from a product state of an arbitrary number of atoms N, dynamics under BOAT leads to the formation of GHZ states for qutrits (d = 3) and ququarts (d = 4). While BOAT could potentially be realized on several platforms where all-to-all coupling is possible, here we propose specific implementations using trapped ion systems. We show that preparing these states with fidelity above a threshold value rules out lower dimensional entanglement than that of the generalized GHZ states. For qutrits, we also propose a protocol to bound the fidelity that requires only global addressing of the ion crystal and single-shot readout of one of the levels. Our results open a path for the scalable generation and certification of high-dimensional multipartite entanglement on current atom-based quantum hardware.

DOI
10.1088/2058-9565/adac06

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Abstract We propose a protocol for the preparation of generalized Greenberger–Horne–Zeilinger (GHZ) states of N atoms each with d = 3 or 4 internal levels. We generalize the celebrated one-axis twisting (OAT) Hamiltonian for N qubits to qudits by including OAT interactions of equal strengths between every pair of qudit levels, a protocol we call as balanced OAT (BOAT). Analogous to OAT for qubits, we find that starting from a product state of an arbitrary number of atoms N, dynamics under BOAT leads to the formation of GHZ states for qutrits (d = 3) and ququarts (d = 4). While BOAT could potentially be realized on several platforms where all-to-all coupling is possible, here we propose specific implementations using trapped ion systems. We show that preparing these states with fidelity above a threshold value rules out lower dimensional entanglement than that of the generalized GHZ states. For qutrits, we also propose a protocol to bound the fidelity that requires only global addressing of the ion crystal and single-shot readout of one of the levels. Our results open a path for the scalable generation and certification of high-dimensional multipartite entanglement on current atom-based quantum hardware.
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.
ABSTRACT High‐dimensional qudit (i.e., ‐level or ‐state) systems outperform 2D qubit (i.e., two‐level or two‐state) systems in some quantum information processing tasks. We exploit entanglement purification protocols (EPPs) for extracting a subset of high‐quality arbitrary ‐dimensional ‐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.
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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.
The Greenberger-Horne-Zeilinger (GHZ) state is a key resource for quantum information processing and quantum metrology. The atomic GHZ state can be generated by one-axis twisting (OAT) interaction H_{OAT}=χJ_{z}^{2} with χ the interaction strength, but it requires a long evolution time χt=π/2 and is thus seriously influenced by decoherence and losses. Here we propose a three-body collective-spin XYZ model which creates a GHZ-like state in a very short timescale χt∼lnN/N for N particles. We show that this model can be effectively produced by applying Floquet driving to an original OAT Hamiltonian. Compared with the ideal GHZ state, the GHZ-like state generated using our model can maintain similar metrological properties reaching the Heisenberg-limited scaling, and it shows better robustness to decoherence and particle losses. This Letter opens the avenue for generating GHZ-like states with a large particle number, which holds great potential for the study of macroscopic quantum effects and for applications in quantum metrology and quantum information.
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.