DeepAI AI Chat
Log In Sign Up

Local Asymptotic Normality and Optimal Estimation of low-rank Quantum Systems

by   Samriddha Lahiry, et al.

In classical statistics, a statistical experiment consisting of n i.i.d observations from d-dimensional multinomial distributions can be well approximated by a d-1 dimensional Gaussian distribution. In a quantum version of the result it has been shown that a collection of n qudits of full rank can be well approximated by a quantum system containing a classical part, which is a d-1 dimensional Gaussian distribution, and a quantum part containing an ensemble of d(d-1)/2 shifted thermal states. In this paper, we obtain a generalization of this result when the qudits are not of full rank. We show that when the rank of the qudits is r, then the limiting experiment consists of an r-1 dimensional Gaussian distribution and an ensemble of both shifted pure and shifted thermal states. We also outline a two-stage procedure for the estimation of the low-rank qudit, where we obtain an estimator which is sharp minimax optimal. For the estimation of a linear functional of the quantum state, we construct an estimator, analyze the risk and use quantum LAN to show that our estimator is also optimal in the minimax sense.


page 1

page 2

page 3

page 4


Low Rank Approximation in Simulations of Quantum Algorithms

Simulating quantum algorithms on classical computers is challenging when...

Estimation of low rank density matrices by Pauli measurements

Density matrices are positively semi-definite Hermitian matrices with un...

The quantum low-rank approximation problem

We consider a quantum version of the famous low-rank approximation probl...

High-dimensional cointegration and Kuramoto systems

This paper presents a novel estimator for a non-standard restriction to ...

Entanglement Properties of Quantum Superpositions of Smooth, Differentiable Functions

We present an entanglement analysis of quantum superpositions correspond...

ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching

In this paper, we develop a novel procedure for low-rank tensor regressi...

Rigorous Guarantees for Tyler's M-estimator via quantum expansion

Estimating the shape of an elliptical distribution is a fundamental prob...