Multimode Cat States in Bosonic Quantum Systems

• Multimode cat states are bosonic superpositions with coherent state branches across several modes, forming effective logical-qubit bases for GHZ-like, cluster, and W-type entanglement.
• They are generated and stabilized via methods such as heralded non-Gaussian measurements, driven Kerr parametric oscillators, and engineered dissipative protocols.
• Advanced tomographic and parity measurements reveal their entanglement and coherence, making them crucial resources for scalable quantum information processing.

Searching arXiv for recent and foundational papers on multimode cat states and closely related bosonic cat resources. Multimode cat states are bosonic superposition states in which coherent-state branches are correlated across several modes, typically as coherent superpositions of branch configurations such as $$\ket{\alpha}^{\otimes M}\pm\ket{-\alpha}^{\otimes M}$$, more general $n$-mode states $\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$, or graph- and code-structured variants defined inside a coherent-branch manifold (Bräuer et al., 3 Jul 2026, Fastovets et al., 2022). In the large-$|\alpha|$ limit, the branches $\ket{\pm\alpha}$ become approximately orthogonal and define an effective logical-qubit basis, which is why multimode cat states are recurrent in GHZ-like entanglement, cluster-state constructions, W-type resources, and rotationally symmetric bosonic codes (Bräuer et al., 3 Jul 2026, Sheng et al., 2019, Hanks et al., 2024).

1. Canonical definitions and state families

A standard $n$-mode entangled coherent state is the equal superposition

$$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$

with

$${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$

For $|\alpha|^2\gg1$, this state approaches a continuous-variable analog of the $n$-qubit GHZ state (Sheng et al., 2019). A more general $n$0-mode Schrödinger cat state is

$n$1

where $n$2 (Fastovets et al., 2022). This form makes explicit that multimode cat states need not be symmetric across modes.

A second major family is the rotationally symmetric cat-code basis $n$3, written either as a superposition of $n$4 coherent states around a circle in phase space or as a Fock-state comb. Here $n$5 is the code distance and half the number of coherent-state components, while $n$6 labels the logical qubit. The code can ideally correct up to $n$7 photon losses (Hanks et al., 2024). This family broadens “multimode cat state” beyond the two-branch GHZ form: the essential feature is coherent support on a structured set of separated quasi-classical branches.

Compact multimode cats also arise in $n$8 and $n$9 settings. For two modes, the binomial cat states are

$\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$0

where the coherent branches lie in a fixed total-excitation manifold $\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$1 and the amplitudes follow binomial weights in the Schwinger representation (Zhao et al., 3 Jul 2026). In this sense, multimode cat states include both non-compact coherent-state superpositions and compact finite-support constructions.

2. Branch manifolds, symmetry sectors, and effective logical structure

A unifying operator-level description begins by confining each oscillator mode to the two-branch coherent support

$\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$2

The multimode branch Hamiltonian

$\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$3

has every branch configuration

$\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$4

in its zero-energy space, so it fixes the coherent-state alphabet but leaves a $\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$5-dimensional degeneracy (Bräuer et al., 3 Jul 2026). State-dependent positive-semidefinite constraints then select the desired correlations and symmetry sector. For GHZ cats, connected alignment terms $\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$6 reduce the branch manifold to $\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$7, and a global parity selector chooses $\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$8. For cluster cats, pair alignment first reduces the branch space and stabilizer-like constraints isolate the unique cluster cat. For W cats, a fixed-defect sector is selected before a noncommuting exchange term produces the equal-amplitude symmetric W superposition (Bräuer et al., 3 Jul 2026).

In the large-$\ket{\mathrm{cat}_{\alpha_1\cdots\alpha_n}}$9 limit, $|\alpha|$0, so

$|\alpha|$1

defines an effective qubit basis. The operator dictionary given in the parent-Hamiltonian construction maps parity to $|\alpha|$2, alignment penalties to Ising-like $|\alpha|$3, exchange to $|\alpha|$4, and global parity to $|\alpha|$5 (Bräuer et al., 3 Jul 2026). This is the direct bridge between coherent-state bosonic engineering and stabilizer-based quantum information processing.

Driven Kerr parametric oscillators provide an allied viewpoint. For a single mode,

$|\alpha|$6

preserves photon-number parity, and $|\alpha|$7 and $|\alpha|$8 are adiabatically connected to even and odd cat states. For two coupled modes, the multimode cat states

$|\alpha|$9

arise as degenerate ground-manifold states, and in the large-amplitude regime the associated proto-Bell cats approach the Bell-cat basis $\ket{\pm\alpha}$0 (Resch et al., 5 May 2025). The structural point is the same in both formulations: multimode cat states are selected from a branch manifold by symmetry, parity, or coupling constraints.

3. Preparation mechanisms

A broad analytic route starts from an arbitrary $\ket{\pm\alpha}$1-mode Gaussian state, measures $\ket{\pm\alpha}$2 modes in the Fock basis, and conditionally prepares an $\ket{\pm\alpha}$3-mode non-Gaussian output. For the single-mode heralding case relevant to cat generation, the output always factorizes as

$\ket{\pm\alpha}$4

so the heralded state is a Gaussian gate acting on a finite Fock superposition (Su et al., 2019). The even cat is a canonical target because it contains only even Fock states. For small $\ket{\pm\alpha}$5, it is well approximated by $\ket{\pm\alpha}$6; for larger $\ket{\pm\alpha}$7, the better ansatz is $\ket{\pm\alpha}$8. In the optimized two-mode realization, Table I reports fidelities above $\ket{\pm\alpha}$9 and success probabilities above $n$0 for $n$1, including

$n$2

with input squeezing $n$3, corresponding to about $n$4–$n$5 dB (Su et al., 2019).

Remote preparation based on a distributed two-mode squeezed state replaces local non-Gaussian synthesis by conditional steering. Alice subtracts photons and performs a homodyne projective measurement on her mode of

$n$6

so that Bob’s conditional state becomes an odd cat-like superposition. The homodyne angle $n$7 remotely rotates the cat, and the loss asymmetry is pronounced: the prepared cat tolerates much more loss in Alice’s channel than in Bob’s channel. The same framework predicts amplitudes larger than $n$8 by increasing the squeezing level and subtracting photon numbers; for three-photon subtraction and squeezing around $n$9 dB, the paper reports amplitude $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$0 and fidelity $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$1 (Han et al., 2023).

Deterministic microwave generation proceeds differently. A transmon qubit dispersively coupled to a 3D cavity reflects a train of coherent microwave pulses so that

$$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$2

and after $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$3 reflections the joint state is

$$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$4

Qubit rotation and projective measurement then prepare even or odd flying multipartite cat states $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$5. Full tomography was demonstrated up to four photonic modes, with reconstructed fidelities $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$6 for the bipartite even cat, $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$7 for the bipartite odd cat, $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$8 for the tripartite even cat, and $$\left|\psi\right\rangle_{n\text{-cat}} = {\cal N}_{n}\left( \left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\cdots \left|\alpha\right\rangle_{n} + \left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2}\cdots \left|-\alpha\right\rangle_{n} \right),$$9 for the quadripartite even cat (Wang et al., 2021).

Resource-efficient optical scaling can also mean preparing several cats at once. A nondegenerate optical parametric amplifier operated in the parametric de-amplification regime generates two squeezed vacuum states with orthogonal squeezing directions in phase space; subtracting one photon from each yields two odd cat states with orthogonal superposition directions, amplitudes about $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$0 and $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$1, fidelities $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$2 and $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$3, Wigner-function negativities at the origin about $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$4 and $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$5, and a generation rate about $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$6 Hz (Han et al., 2023). This is a distinct scaling notion from a single entangled multimode cat, but it is directly relevant to multi-cat-state architectures.

4. Tomography, phase-space structure, and multipartite nonclassical correlations

The Wigner representation remains central, but multimode cat states admit a complete positive-probability description through center-of-mass tomography. For $${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$7 degrees of freedom, the center-of-mass tomogram

$${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$8

is nonnegative, normalized, and related to the Wigner function by the Radon-type projection

$${\cal N}_{n}=\left[2+2e^{-2n|\alpha|^{2}}\right]^{-1/2}.$$9

For a two-mode cat state, the explicit tomogram contains two displaced Gaussian contributions and interference cross terms, and the same formalism yields the linear entropy

$|\alpha|^2\gg1$0

as an entanglement diagnostic (Dudinets et al., 2018).

Experimental itinerant-state tomography of flying multipartite cats is based on complex-amplitude histograms $|\alpha|^2\gg1$1 of the outgoing field and maximum-likelihood reconstruction of the full $|\alpha|^2\gg1$2-dimensional histogram $|\alpha|^2\gg1$3. In the reported circuit-QED experiment, the reconstructed density matrices show dominant populations in the coherent-state subspace, with cat-basis subspace traces about $|\alpha|^2\gg1$4 and $|\alpha|^2\gg1$5 for single-mode even and odd cats and about $|\alpha|^2\gg1$6 and $|\alpha|^2\gg1$7 for quadripartite even and odd cats. Multipartite entanglement was certified by localizable entanglement, and hybrid qubit-cat entanglement by negativities

$|\alpha|^2\gg1$8

for qubit–single-cat and qubit–bipartite-cat systems (Wang et al., 2021).

For Bell-type nonlocality, rotated quantum-number parity measurements are especially effective. Using

$|\alpha|^2\gg1$9

the Mermin-Klyshko signal can approach the maximal quantum value already when the average quantum number per mode $n$0 is only about $n$1. For the three-mode cat, $n$2 reaches $n$3 at $n$4, close to the upper bound $n$5; for four and five modes, the rescaled signals at $n$6 are $n$7 and $n$8, and the inequality violation grows exponentially with the number of entangled modes (Sheng et al., 2019).

5. Decoherence, loss, numerical simulation, and stabilization

A multimode cat can be reduced to an effective two-mode interfering-state model by partitioning modes into a “system” block and an “environment” block. In that reduction, the Schmidt number

$n$9

and the visibility

$n$00

are linked by

$n$01

For the identical-mode case $n$02, the paper gives the explicit multimode-cat visibility

$n$03

where $n$04 is the number of environment modes measured or lost (Fastovets et al., 2022). The same analysis introduces the “health” parameter

$n$05

which quantifies how sequential conditioning breaks the balance between the two macroscopic branches.

The asymptotic size of a cat does not, by itself, force the nonclassical distance toward its maximum. For single-mode even and odd Schrödinger cats,

$n$06

obeys bounds determined by the Husimi function, and both even and odd cat states satisfy $n$07 as $n$08. The paper explicitly emphasizes that “the nonclassical distance of the cat states is bounded away from unity regardless of the superposition amplitude,” and that nonclassical distance is not necessarily monotonically increasing with respect to macroscopicity (Nair, 2017). This is a recurrent corrective to the intuition that wider phase-space separation always implies “more nonclassicality.”

Large-array dynamics requires scalable numerics. In a driven-dissipative chain with two-photon drive, one-photon loss, two-photon loss, and engineered nonlocal dissipation, the positive-$n$09 representation yields exact stochastic differential equations for multimode cat-state transients and supports simulations up to $n$10 sites, with the paper noting scalability to $n$11. The main limitation is parity: the estimator

$n$12

is exponentially sensitive to rare trajectories and becomes numerically unstable even when photon-number and correlation observables remain accurate (Shi et al., 11 Jan 2026).

Autonomous protection can also be designed directly. Dissipative preparation and stabilization of compact multimode cats is achieved by multiple Lindblad jump operators whose common kernel is the target state manifold. For the two-mode binomial cat, the proposed operators

$n$13

pump the system from vacuum into the fixed-$n$14 manifold and then lock the cat structure inside it. The reported effect is to prepare the target from vacuum with high fidelity and to extend its lifetime by several orders of magnitude compared to natural decay times (Zhao et al., 3 Jul 2026).

6. Quantum-information roles, protocol variants, and conceptual boundaries

In bosonic error correction, multimode cat resources appear both as logical states and as ancillae. Teleportation-based correction for rotationally symmetric codes uses controlled-rotation gates

$n$15

and phase measurements, but the ancillary state need not be an exact logical $n$16 cat. Many-component Yurke–Stoler states, although not themselves in the cat-code space, can still function as “bridge” states because known extra rotations $n$17 and known photon-loss events $n$18 can be absorbed into the classical decoding rules (Hanks et al., 2024). This broadens the operational notion of a useful multimode cat resource.

Distributed-network applications use mode multiplexing rather than only local multimode entanglement. In the quantum-repeater proposal based on entangled phase-modulated multimode coherent states, electro-optic modulation transforms a one-mode cat into a frequency-comb state

$n$19

and interference of selected subcarrier modes at a central beam splitter, followed by photon counting, heralds entanglement between remote memories (Goncharov et al., 2022). A related three-mode teleportation protocol builds an entangled mode pair by beam-splitting a cat state and then teleports an unknown cat-encoded qubit using beam splitters, parity or joint-parity measurements, and a final binary-outcome dispersive readout; with multiple measurements, the protocol can approach unit fidelity (Feng et al., 2024).

Two conceptual boundaries are important. First, not every “cat-like” structured state is a canonical even or odd cat. In the doubly pumped intraresonance Kerr microresonator, the reduced $n$20 resonant photon-conversion model exhibits a residual $n$21 symmetry and produces multicomponent Schrödinger-cat-like states characterized by Wigner negativity, non-Poissonian statistics, pump-mode quadrature squeezing, and large single-mode Schmidt numbers, but the paper explicitly states that these are not the canonical even or odd coherent states of Dodonov, Malkin, and Man’ko (Singh et al., 17 Jun 2026). Second, classical multimode analogs exist but are not quantum cat states. The OAM-based “analogous cat state” is a coherent superposition over many orbital-angular-momentum modes with high similarity to theory in state tomography, yet the paper explicitly notes that it has zero quantum macroscopicity because it is made of classical electromagnetic modes (Liu et al., 2018).

Taken together, these constructions show that “multimode cat state” is best understood as a family of structured bosonic superpositions defined by coherent-branch support, intermode correlation constraints, and symmetry-sector selection. In current usage, the term covers GHZ-like entangled coherent states, compact multinomial and binomial cats, distributed frequency-multiplexed resources, Bell-cat states in coupled Kerr systems, and fault-tolerant ancillary states that retain the relevant syndrome-extraction structure even when they are not exact codewords (Bräuer et al., 3 Jul 2026, Resch et al., 5 May 2025, Hanks et al., 2024).