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arXiv:2406.17999 · Nature Communications

Entangling Schrödinger’s cat states by bridging discrete- and continuous-variable encoding

Daisuke Hoshi, Toshiaki Nagase, Sangil Kwon, Daisuke Iyama, Takahiko Kamiya, Shiori Fujii, Hiroto Mukai, Shahnawaz Ahmed, Anton Frisk Kockum, Shohei Watabe, Fumiki Yoshihara, Jaw-Shen Tsai

Published Feb 3, 2025• 16 citations
arXivHTMLPDF

Abstract

In quantum information processing, two primary research directions have emerged: one based on discrete variables (DV) and the other on the structure of quantum states in a continuous-variable (CV) space. Integrating these two approaches could unlock new potentials, overcoming their respective limitations. Here, we show that such a DV–CV hybrid approach, applied to superconducting Kerr parametric oscillators (KPOs), enables us to entangle a pair of Schrödinger’s cat states by two methods. The first involves the entanglement-preserving conversion between Bell states in the Fock-state basis (DV encoding) and those in the cat-state basis (CV encoding). The second method implements a $$\sqrt{{{{\rm{iSWAP}}}}}$$ gate between two cat states following the procedure for Fock-state encoding. This simple and fast gate operation completes a universal quantum gate set in a KPO system. Our work offers powerful applications of DV–CV hybridization and marks a first step toward developing a multi-qubit platform based on planar KPO systems. Quantum information processing normally uses either discrete- or continuous-variable encoding. Here, the authors bridge the two approaches, showing how to entangle Schrodinger’s cats states by conversion between Bell states in the Fock and cat bases and by a simple Fock-state-like gate operation.

DOI
10.1038/s41467-025-56503-8

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In quantum information processing, two primary research directions have emerged: one based on discrete variables (DV) and the other on the structure of quantum states in a continuous-variable (CV) space. Integrating these two approaches could unlock new potentials, overcoming their respective limitations. Here, we show that such a DV–CV hybrid approach, applied to superconducting Kerr parametric oscillators (KPOs), enables us to entangle a pair of Schrödinger’s cat states by two methods. The first involves the entanglement-preserving conversion between Bell states in the Fock-state basis (DV encoding) and those in the cat-state basis (CV encoding). The second method implements a iSWAP\sqrt{{{{\rm{iSWAP}}}}}iSWAP​ gate between two cat states following the procedure for Fock-state encoding. This simple and fast gate operation completes a universal quantum gate set in a KPO system. Our work offers powerful applications of DV–CV hybridization and marks a first step toward developing a multi-qubit platform based on planar KPO systems. Quantum information processing normally uses either discrete- or continuous-variable encoding. Here, the authors bridge the two approaches, showing how to entangle Schrodinger’s cats states by conversion between Bell states in the Fock and cat bases and by a simple Fock-state-like gate operation.
We show that a continuous-variable (CV) qudit can be constructed using quasiorthogonal cat states of a bosonic mode, when the phase encoded in each cat state is chosen appropriately. With the constructed CV qudit and the discrete-variable (DV) qudit encoded with Fock states, we propose an approach to generate the hybrid maximally entangled state of a CV qudit and a DV qudit by using two microwave cavities coupled to a superconducting flux qutrit. This proposal relies on the initial preparation of a superposition of Fock states of one cavity and the initial preparation of a cat state of the other cavity. After the initial state of each cavity is prepared, this proposal requires only two basic operations, i.e., the first operation employs the dispersive coupling of both cavities with the qutrit while the second operation uses the dispersive coupling of only one cavity with the qutrit. The entangled state production is deterministic and the operation time decreases as the dimensional size of each qudit increases. In addition, during the entire operation, the coupler qutrit remains in the ground state and thus decoherence from the qutrit is significantly reduced. As an example, we further discuss the experimental feasibility for generating the hybrid maximally entangled state of a DV qutrit and a CV qutrit based on circuit QED. This proposal is universal and can be extended to accomplish the same task, by using two microwave or optical cavities coupled to a natural or artificial three-level atom.
Quantum technologies are rapidly emerging as a foundation for next-generation information processing, communication, and sensing. In particular, optical platforms stand out as a promising avenue for scalable and efficient architectures. Here, information can be encoded either in discrete variables (DV)—such as polarization, path, frequency, or angular momentum—or in continuous variables (CV) of the optical field, such as quadratures. We develope a new formalism for the joint description of multiphoton and multimode light undergoing Gaussian transformations, and we use it to explore two DV-CV hybrid protocols. The first unifies and extends distinct forms of boson sampling, merging DV scattershot boson sampling with CV Gaussian boson sampling, thereby enabling quantum advantage demonstrations with squeezed photons. The second protocol deals with the Bell state measurement (BSM) for qudits, which are a higher-dimensional generalization of the qubit. Bell measurements play a key role in fusion-based quantum computation entanglement distribution and secret key sharing. We therefore introduce high-dimensional BSMs through linear interferometry combined with multiple single-mode squeezers, outperforming state-of-the-art theoretical methods in linear optics without using auxiliary states. This result shows how bridging the DV and CV paradigms provides a framework in which their combined strengths can outperform their standalone applications and overcome their individual limitations.
Schrödinger cat states play an important role for applications in continuous variable quantum information technologies. As macroscopic superpositions, they are inherently protected against certain types of noise making cat qubits a promising candidate for quantum computing. It has been shown recently that cat states occur naturally in driven Kerr parametric oscillators as degenerate ground states with even and odd parity that are adiabatically connected to the respective lowest two Fock states by switching off the drive. To perform operations with several cat qubits, one crucial task is to create entanglement between them. Here, we demonstrate efficient transformations of multimode cat states through adiabatic and diabatic switching between Kerr-type Hamiltonians with degenerate ground-state manifolds. These transformations can be used to directly initialize the cats as entangled Bell states in contrast to initializing them from entangled Fock states.
In optical quantum information, one can distinguish two approaches: the discrete-variable (DV), involving single photons, for example c0|0〉 + c1|1〉, and the continuous-variable alternative (CV) encoding information into the quadrature components of the light field. In this CV approach, qubits can be implemented as superposition of coherent states of opposite phases, |cat ±〉 = |α〉 ± |−α〉, also called optical Schrodinger cat states. Combining these two approaches has recently led to a so-called optical hybrid quantum information aiming at merging the properties of both [1]. As both encodings have their own advantages and drawbacks, transferring information between them is a requirement which can be fulfill by teleportation using hybrid entanglement between particle-like qubits and wave-like qubits, i.e. entanglement of the form |0〉|α〉 + |1〉|−α〉.
Microwave-optical interactions and hybrid entangled states are crucial for building hybrid quantum networks. In this work, we propose an approach for realizing a cross-Kerr nonlinear interaction between microwave and optical cavities using an ensemble of nitrogen-vacancy centers (NV ensemble). This cross-Kerr interaction is achieved by an NV ensemble dispersively coupled to a microwave cavity and an optical cavity. As an application, we show that the cross-Kerr interaction can be used to create a hybrid entangled state of a discrete-variable (DV) optical qubit and a continuous-variable (CV) optical qubit. The DV optical qubit here refers to a qubit, with two logic states encoded via the vacuum and single-photon states of the optical cavity. The CV optical qubit means a qubit, whose two logic states are encoded through two quasi-orthogonal coherent states or cat states of the microwave cavity. Our method is quite simple because it only requires a single-step operation. Numerical simulations demonstrate that high-fidelity generation of the proposed hybrid state is feasible within current experimental technology. This proposal is universal and the NV ensemble can be replaced by other quantum transducers, such as magnons, rare-earth-doped crystals, and silicon-vacancy centers.
Transferring quantum information between different types of quantum hardware is crucial for integrated quantum technology. In particular, converting information between continuous variable (CV) and discrete variable (DV) devices enables many applications in quantum networking, quantum sensing, quantum machine learning, and quantum computing. This paper addresses the transfer of CV-encoded information between CV and DV devices. We present a resource-efficient method for encoding CV states and implementing CV gates on DV devices, as well as two measurement-based protocols for transferring CV states between CV and DV devices. The success probability of the transfer protocols depends on the measurement outcome and can be increased to near-deterministic values by adding ancillary qubits to the DV devices.