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Asymptotic inference in a stationary quantum time series

Michael Nussbaum, Arleta Szkoła

TL;DR

The paper develops a quantum analogue of classical spectral-density inference for n-mode Gaussian time series by modeling shift- and gauge-invariant quantum states via Toeplitz-symbols and quantum Le Cam distance. It proves that the quantum experiment based on the spectral-density parameter is asymptotically Δ-equivalent to a classical geometric-regression model, which in turn is asymptotically equivalent to a Gaussian white-noise model after an arc-cosh type transformation of the spectral density. The key contributions include a detailed upper and lower informativity analysis, circulant-Toeplitz symbol approximations, and construction of unbiased and one-step efficient estimators for the quantum spectral density, establishing a comprehensive quantum-to-classical asymptotic equivalence framework. This work lays the groundwork for quantum spectral-density estimation and suggests quantum analogues of classical tools like the periodogram, with potential extensions to optimal parametric and nonparametric estimation in the quantum setting.

Abstract

We consider a statistical model of a n-mode quantum Gaussian state which is shift invariant and also gauge invariant. Such models can be considered analogs of classical Gaussian stationary time series, parametrized by their spectral density. Defining an appropriate quantum spectral density as the parameter, we establish that the quantum Gaussian time series model is asymptotically equivalent to a classical nonlinear regression model given as a collection of independent geometric random variables. The asymptotic equivalence is established in the sense of the quantum Le Cam distance between statistical models (experiments). The geometric regression model has a further classical approximation as a certain Gaussian white noise model with a transformed quantum spectral density as signal. In this sense, the result is a quantum analog of the asymptotic equivalence of classical spectral density estimation and Gaussian white noise, which is known for Gaussian stationary time series. In a forthcoming version of this preprint, we will also identify a quantum analog of the periodogram and provide optimal parametric and nonparametric estimates of the quantum spectral density.

Asymptotic inference in a stationary quantum time series

TL;DR

The paper develops a quantum analogue of classical spectral-density inference for n-mode Gaussian time series by modeling shift- and gauge-invariant quantum states via Toeplitz-symbols and quantum Le Cam distance. It proves that the quantum experiment based on the spectral-density parameter is asymptotically Δ-equivalent to a classical geometric-regression model, which in turn is asymptotically equivalent to a Gaussian white-noise model after an arc-cosh type transformation of the spectral density. The key contributions include a detailed upper and lower informativity analysis, circulant-Toeplitz symbol approximations, and construction of unbiased and one-step efficient estimators for the quantum spectral density, establishing a comprehensive quantum-to-classical asymptotic equivalence framework. This work lays the groundwork for quantum spectral-density estimation and suggests quantum analogues of classical tools like the periodogram, with potential extensions to optimal parametric and nonparametric estimation in the quantum setting.

Abstract

We consider a statistical model of a n-mode quantum Gaussian state which is shift invariant and also gauge invariant. Such models can be considered analogs of classical Gaussian stationary time series, parametrized by their spectral density. Defining an appropriate quantum spectral density as the parameter, we establish that the quantum Gaussian time series model is asymptotically equivalent to a classical nonlinear regression model given as a collection of independent geometric random variables. The asymptotic equivalence is established in the sense of the quantum Le Cam distance between statistical models (experiments). The geometric regression model has a further classical approximation as a certain Gaussian white noise model with a transformed quantum spectral density as signal. In this sense, the result is a quantum analog of the asymptotic equivalence of classical spectral density estimation and Gaussian white noise, which is known for Gaussian stationary time series. In a forthcoming version of this preprint, we will also identify a quantum analog of the periodogram and provide optimal parametric and nonparametric estimates of the quantum spectral density.

Paper Structure

This paper contains 63 sections, 64 theorems, 964 equations.

Table of Contents

  1. Main Results
  2. Introduction
  3. Gaussian states
  4. Shift invariant states
  5. Gauge invariant states
  6. The asymptotic setup
  7. Quantum Le Cam distance
  8. Main theorems
  9. Discussion
  10. Upper informativity bound
  11. Gaussian states on symmetric Fock space
  12. Distance of states in terms of symbols
  13. Approximation of Toeplitz matrices
  14. Upper information bound via approximation of symbols
  15. The geometric regression model
  16. ...and 48 more sections

Key Result

Theorem 1.1

If $\Theta=\Theta_{1}\left( \alpha,M\right)$ for some $\alpha>1/2,$$M>1$ then i.e. $\mathcal{F}_{n}\left( \Theta\right)$ is asymptotically more informative than $\mathcal{E}_{n}\left( \Theta\right)$: $\mathcal{E}_{n}\left( \Theta\right) \lesssim\mathcal{F}_{n}\left( \Theta\right)$.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • ...and 57 more