Nonparametric Learning Non-Gaussian Quantum States of Continuous Variable Systems

Abstract

Continuous-variable quantum systems are foundational to quantum computation, communication, and sensing. While traditional representations using wave functions or density matrices are often impractical, the tomographic picture of quantum mechanics provides an accessible alternative by associating quantum states with classical probability distribution functions called tomograms. Despite its advantages, including compatibility with classical statistical methods, tomographic method remain underutilized due to a lack of robust estimation techniques. This work addresses this gap by introducing a non-parametric \emph{kernel quantum state estimation} (KQSE) framework for reconstructing quantum states and their trace characteristics from noisy data, without prior knowledge of the state. In contrast to existing methods, KQSE yields estimates of the density matrix in various bases, as well as trace quantities such as purity, higher moments, overlap, and trace distance, with a near-optimal convergence rate of $\tilde{O}\bigl(T^{-1}\bigr)$, where $T$ is the total number of measurements. KQSE is robust for multimodal, non-Gaussian states, making it particularly well suited for characterizing states essential for quantum science.

Figures (21)

• Figure 1: In the left panel, a schematic diagram shows how the density operator $\boldsymbol{\rho}$, the tomogram $\mathcal{W}(x| \mu,\nu)$, the Wigner function $W(q,p)$, and the CF $\phi(1;\mu,\nu)$ are related. These representations are equivalent and interconvertible via one‐to‐one mappings. In the right panel, the flowchart depicts how each representation is reconstructed from the measured data $\{X_1,\dots,X_n\}_{\mu,\nu}$. Here, hats denote estimates rather than operators (e.g. $\widehat{\mathcal{W}}(x| \mu,\nu)$ is an estimate of $\mathcal{W}(x| \mu,\nu)$). In particular, the Wigner function can be obtained from either the tomogram or the CF.
• Figure 2: Comparison of CCS tomogram reconstructions using MLE ($GMM=2,3$) and Gaussian KDE for $n=1000$ with $\mu=0.8$, $\nu=1.2$, and $a=1+0.5i$.
• Figure 3: Comparison of CCS CF estimator obtained using MLE ($GMM=2,3$) and KCFE for $n=1000$ and $\nu=1.2$.
• Figure 4: Comparison of estimation errors with their theoretical upper bounds for the initial coherent cat state (a) Comparison between $\varepsilon_K^2$ (Eq. \ref{['234']}) and its upper bounds (Eqs. \ref{['234_1']} and \ref{['234_2']} for upper bound 1 and 2 in the labels, respectively) as a function of $n\in[100,2000]$ for $N_{\mu}=50$, $\mu_{\max}=6$, $(y,y')=(1.5,1.0)$. Each point is averaged over 100 trials. Observed ratios: $46.07$ and $55.37$. (b) Comparison between $\varepsilon_{\rho}$ (Eq. \ref{['1033_33']}) and its upper bound (Eq. \ref{['eq:epsilon_rho_upperbound']}) as a function of $T\in [10^4, 10^5]$.The observed ratio is $4.88 \times 10^2$.
• Figure 5: Comparison of KDEs and histogram with the CCS tomogram (PDF) \ref{['1328']} for different sample sizes $n=\{500,1000,2000\}$ for a fixed $\mu = 0.8$, $\nu = 1.2$, $a = 1.0 + 0.5i$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• proof
• Example 1
• Example 2