Optimal estimation of qubit states with continuous time measurements

Abstract

We propose an adaptive, two steps strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given $n$ identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large $n$, the statistical model described by $n$ identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term `local' refers to a shrinking neighborhood around a fixed state ${\rho }_0$. An essential result is that the neighborhood radius can be chosen arbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius $n^{-1/2+\epsilon }$, and then use LAN to perform optimal estimation.

32 pages, 3 figures, to appear in Commun. Math. Phys

References in corpus (3)

• Local asymptotic normality in quantum statistics
• Asymptotic performance of optimal state estimation in quantum two level system
• Quantum estimation and the quantum central limit theorem

Cited by in corpus (27)

• Multi-parameter Quantum Metrology
• Multi-parameter estimation beyond Quantum Fisher Information
• Sample-optimal tomography of quantum states
• Local asymptotic normality for finite dimensional quantum systems
• Attaining the ultimate precision limit in quantum state estimation
• On Asymptotic Quantum Statistical Inference
• Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation
• Noisy quantum metrology enhanced by continuous nondemolition measurement
• Local asymptotic normality in quantum statistics
• Fisher information and asymptotic normality in system identification for quantum Markov chains
• Quantum learning: optimal classification of qubit states
• Daemonic ergotropy in continuously-monitored open quantum batteries
• Fourier Analytic Approach to Phase Estimation
• Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics
• Asymptotic inference in system identification for the atom maser
• Experimental simulation of daemonic work extraction in open quantum batteries on a digital quantum computer
• Quantum teleportation benchmarks for independent and identically-distributed spin states and displaced thermal states
• Sample efficient tomography via Pauli Measurements
• Asymptotically optimal purification and dilution of mixed qubit and Gaussian states
• Local asymptotic equivalence of pure quantum states ensembles and quantum Gaussian white noise
• Universal Optimal Estimation of the Polarization of Light with Arbitrary Photon Statistics
• Noise-assisted and monitoring-enhanced quantum bath tagging
• Quantum U-statistics
• Minimax estimation of qubit states with Bures risk
• Topics in Estimation of Quantum Channels
• Asymptotically optimal quantum channel reversal for qudit ensembles and multimode Gaussian states
• Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimator