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arXiv:2607.02997

Symmetry-Resolved Parent Hamiltonians for Entangled Bosonic Cat Resources

Šimon Bräuer, Klaus Mølmer

quant-phSubmitted Jul 3, 2026• Announced Jul 6, 2026• 0 citations
v1 · Jul 3, 2026v2 · Oct 1, 2026
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Abstract

We derive parent Hamiltonians in terms of oscillator operators for multimode bosonic cat resource states. The construction separates a universal branch Hamiltonian, which confines each mode to the coherent-state support $ |\pmα\rangle$, and state-dependent constraint Hamiltonians, which select the desired correlations and symmetry sectors inside the resulting branch manifold. This framework progressively removes degeneracies in the low-energy manifold and yields explicit parent Hamiltonians for GHZ-, cluster-, and W-type cat states. In the large-$|α|$ limit, the bosonic parent Hamiltonians reduce to stabilizer or exchange Hamiltonians acting on an effective logical-qubit basis. The present construction provides a direct bridge between coherent-state bosonic engineering and stabilizer-based quantum information processing.

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We derive parent Hamiltonians in terms of oscillator operators for multimode bosonic cat resource states. The construction separates a universal branch Hamiltonian, which confines each mode to the coherent-state support ∣±α⟩ |\pmα\rangle∣±α⟩, and state-dependent constraint Hamiltonians, which select the desired correlations and symmetry sectors inside the resulting branch manifold. This framework progressively removes degeneracies in the low-energy manifold and yields explicit parent Hamiltonians for GHZ-, cluster-, and W-type cat states. In the large-∣α∣|α|∣α∣ limit, the bosonic parent Hamiltonians reduce to stabilizer or exchange Hamiltonians acting on an effective logical-qubit basis. The present construction provides a direct bridge between coherent-state bosonic engineering and stabilizer-based quantum information processing.
We motivate and introduce a class of “hierarchical” quantum superposition states of N coupled quantum oscillators. Unlike other well-known multimode photonic Schrödinger-cat states such as entangled coherent states, the hierarchical superposition states are characterized as two-branch superpositions of tensor products of single-mode Schrödinger-cat states. In addition to analyzing the photon statistics and quasiprobability distributions of prominent examples of these nonclassical states, we consider their usefulness for highprecision quantum metrology of nonlinear optical Hamiltonians and quantify their mode entanglement. We propose two methods for generating hierarchical superpositions in N = 2 coupled microwave cavities, exploiting currently existing quantum optical technology for generating entanglement between spatially separated electromagnetic field modes.
Overcomplete families of states of the type of Barut-Girardello coherent states (BG CS) are constructed for noncompact algebras u(p,q)u(p,q)u(p,q) and sp(N,C)sp(N,C)sp(N,C) in quadratic bosonic representation. The sp(N,C)sp(N,C)sp(N,C) BG CS are obtained in the form of multimode ordinary Schrödinger cat states. A set of such macroscopic superpositions is pointed out which is overcomplete in the whole NNN mode Hilbert space (while the associated sp(N,C)sp(N,C)sp(N,C) representation is reducible). The multimode squared amplitude Schrödinger cat states are introduced as macroscopic superpositions of the obtained sp(N,C)sp(N,C)sp(N,C) BG CS.}
Schrödinger cat states provide a hardware-efficient platform for bosonic quantum error correction by encoding logical information in protected manifolds of harmonic oscillators. While previous work has demonstrated the dissipative stabilization of multi-mode Schrödinger cat states as robust quantum memories, a framework for universal quantum computation has remained unavailable. Here we work towards this by introducing a universal gate set for dissipatively stabilized multi-mode cat qubits as a necessary resource for computation. Using a chain of Kerr non-linear oscillators coupled through engineered non-local dissipation and an effective low-dimensional description, we show how single-qubit control can be achieved through arbitrary rotation around the XXX-axis and π/2π/2π/2-rotation around the ZZZ-axis. We further show how coupling two such stabilized arrays through just one oscillator on each respective array enables coherent entangling operations through implementation of the XXπ/2XX_{π/2}XXπ/2​ gate. Numerical simulations demonstrate high-fidelity gate dynamics and entanglement generation under realistic parameters. Finally, we analyze the effects of induced and intrinsic photon loss, disorder, and the validity regime of the effective low-dimensional theory. Our results establish dissipatively stabilized multi-mode Schrödinger cat states as a potential architecture for universal bosonic quantum computation.
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.
Superpositions of macroscopically distinct quantum states, introduced in Schrödinger’s famous Gedankenexperiment, are an epitome of quantum “strangeness” and a natural tool for determining the validity limits of quantum physics. The optical incarnation of Schrödinger’s cat — the superposition of two opposite-amplitude coherent states — is also the backbone of quantum information processing in the continuous-variable domain. Here we implement two protocols for producing and amplifying optical Schrödinger’s cats. In the first protocol we remotely prepare a high-efficiency Schrödinger cat state by applying the remote state preparation to a N00N state prepared between two parties that are separated by a lossy medium. The second protocol consists in bringing the initial states into interference on a beamsplitter and a subsequent heralding quadrature measurement in one of the output channels. The latter technique enables implementation in an iterative manner, in principle allowing creation of Schrödinger’s cat states of arbitrarily high amplitude.
Schrödinger cat states, representing coherent superpositions of macroscopically distinguishable states, are indispensable nonclassical resources for continuous-variable quantum information processing. Existing generation protocols typically rely on strong nonlinear interactions, complicated control techniques, or engineered dissipation, posing challenges for experimental implementation. Here, we propose a simple measurement-based protocol for generating Schrödinger cat states in a cavity-QED system by combining coherent driving, dispersive atom--cavity interactions, and atomic postselection. The atom--cavity interaction establishes coherent correlations between the atomic and photonic degrees of freedom, while the subsequent atomic postselection projects the cavity field onto a non-Gaussian superposition state with pronounced Wigner negativity. Numerical simulations based on the Lindblad master equation show that the generated Schrödinger cat states remain robust against moderate cavity dissipation. Our results demonstrate that conditional atomic measurements provide an effective and experimentally accessible approach for preparing nonclassical cavity states without relying on strong optical nonlinearities or engineered dissipation.