Normalizing Flow-Based Bayesian Parameter Estimation for Noisy Quantum States

Quantum state reconstruction typically focuses on finding a single best-fit density matrix, which often obscures the underlying physical mechanisms and noise sources in an experiment. This paper asks whether it is possible to perform efficient Bayesian parameter estimation (BPE) directly on physically meaningful parameters—such as squeezing strength, thermal noise, and photon-addition fractions—rather than reconstructing the full density matrix. The goal is to provide experimentalists with a probabilistic understanding of how specific physical processes contribute to the observed quantum state.

The authors develop a framework that combines variational inference with normalizing flows, a type of neural network capable of learning complex probability distributions. By embedding a physically interpretable parameter vector into a quantum state ansatz, the model maps simple Gaussian latent variables to the constrained space of physical parameters. The framework uses a two-stage Bayesian strategy: a calibration subset of the data is used to construct a weakly informative prior, while the remaining inference data is used to calculate the likelihood. This separation prevents the same data from being used twice, ensuring a robust posterior distribution. The approach is demonstrated on optical cat states generated via photon-addition, comparing two different physical models (ansatzes) to determine which better explains the experimental data.

The framework successfully resolves the contributions of multiple physical mechanisms, such as photon-addition fractions and thermal noise, from homodyne measurement data. By analyzing the joint posterior distributions, the authors identify strong correlations between specific physical parameters and quantum features like Wigner function negativity. For instance, they observe a consistent anticorrelation between the photon-addition fraction and the Wigner function at the origin, providing a clear, interpretable link between experimental settings and the resulting non-classicality of the state. The method is computationally efficient enough to compare multiple competing physical models, allowing researchers to systematically evaluate which experimental hypotheses best match their observations.