Superpositions of Quantum Gaussian Processes

TL;DR

This work presents Gaussian-Branched Cat States (GCS) as a phase-space description of hybrid CV–DV systems, where each branch corresponds to a Gaussian CV state labeled by a qubit configuration. It derives closed-form, operator-valued Gaussian dynamics in terms of evolving branch phase-space quantities (first moments $r_{jk}$, covariances $\sigma_{jk}$, and QRDM exponents $r^{(0)}_{jk}$), enabling exact unitary and Markovian open evolution as well as Gaussian measurements without Hilbert-space truncation. The authors generalize the framework to $N$ qubits and include diffusion/dephasing noise, providing a scalable set of ODEs (and integral forms in the linear-coupling case) for the full dynamics and measurement statistics. Through two physical examples—measurement-based entanglement with a squeezed resonator and Stern-Gerlach interferometry of a diffusing levitated nanoparticle—the paper demonstrates how GCSs encode non-Gaussian features (coherence between branches and Wigner negativity) in a controllable, experimentally relevant setting. The approach offers a versatile, exact tool for predicting and engineering non-Gaussian quantum processes in CV–DV hybrids with potential impacts on quantum information processing and foundational tests.

Abstract

We generalise the Gaussian formalism of Continuous Variable (CV) systems to describe their interactions with qubits/qudits that result in quantum superpositions of Gaussian processes. To this end, we derive a new set of equations in closed form, which allows us to treat hybrid systems' unitary and open dynamics exactly (without truncation), as well as measurements (ideal and noisy). The $N$-qubits $n$-modes entangled states arising during such processes are named Gaussian-Branched Cat States (GCSs). They are fully characterised by their superposed phase-space quantities: sets of generalised complex first moments and covariance matrices, along with the qubit reduced density matrix (QRDM). We showcase our general formalism with two paradigmatic examples: i) measurement-based entanglement of two qubits via a squeezed, leaking, and measured resonator; ii) the generation of the Wigner negativity of a levitated nanoparticle undergoing Stern-Gerlach interferometry in a diffusive environment.

Figures (4)

• Figure 1: Schematic representation of an initial Gaussian State and a qubit which undergoes a superposition of Gaussian processes, generating a GCS.
• Figure 2: Schematic Representations of the two presented examples along with the Wigner functions of the related statistical mixture of Gaussian processes: (a) Resonator entangled via frequency shift $\chi$ to two qubits undergoing measurement-based entanglement ($\eta$, homodyne efficiency and $\kappa$, decay rate) and (b) a Stern-Gerlach matter-wave interferometer of a mass $m$ trapped at frequency $\omega$ with operator valued force $f_q$ in diffusive environment ($\Gamma_x$).
• Figure 3: Measurement based entanglement between two qubits with a squeezed resonator undergoing momentum homodyne measurement ($\chi = 1$, $\kappa = 3$, $x_0 = 20$, $\eta = 0.6$) for different squeezing parameters $s$: (a) time evolution of the measurement uncertainties, (b) probability distribution at the maximum superposition ($\tau_{\text{max}}$), and (c) conditional negativity of the post measurement state at $\tau_{\text{max}}$.
• Figure 4: $\sigma_x$ measurement of a Stern-Gerlach Interferometry with a hypothetically unknown force ($f_u = 2$, $f_q = 0.5$): (a) Decay in contrast ($\mathcal{C}$) as a function of time ($\tau$) for diffusive ($\Gamma_x = 0.1$), dephasing ($\Gamma_z = 0.8$), and unitary dynamics with initial coherent state, thermal state ($N_p = 0.8$) and thermal squeezed state ($N_p = 0.8$, $s=2$); (b) Wigner functions of the two post-measurement states ($\tau = \pi$, $N_p = 0.8$, $s=2$, $\Gamma_x = 0.02$) .