Local asymptotic normality for finite dimensional quantum systems

TL;DR

This paper generalizes local asymptotic normality (LAN) to finite-dimensional quantum systems, enabling optimal estimation strategies for unknown quantum states by approximating large systems with classical and quantum Gaussian models.

Contribution

It extends LAN results from qubits to arbitrary finite-dimensional quantum systems, providing a foundation for optimal quantum state estimation.

Findings

LAN holds for all finite dimensions $d$, with the limit model combining classical and quantum Gaussian variables.

The limit model separates diagonal and off-diagonal parameters into classical and quantum components.

This framework facilitates the development of optimal adaptive measurement procedures for quantum state estimation.

Abstract

We extend our previous results on local asymptotic normality (LAN) for qubits, to quantum systems of arbitrary finite dimension $d$. LAN means that the quantum statistical model consisting of $n$ identically prepared $d$-dimensional systems with joint state $\rho^{\otimes n}$ converges as $n\to\infty$ to a statistical model consisting of classical and quantum Gaussian variables with fixed and known covariance matrix, and unknown means related to the parameters of the density matrix $\rho$. Remarkably, the limit model splits into a product of a classical Gaussian with mean equal to the diagonal parameters, and independent harmonic oscillators prepared in thermal equilibrium states displaced by an amount proportional to the off-diagonal elements. As in the qubits case, LAN is the main ingredient in devising a general two step adaptive procedure for the optimal estimation of completely unknown $d$-dimensional quantum states. This measurement strategy shall be described in a forthcoming paper.