Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory

TL;DR

This paper investigates quantum tomography using random diagonal unitaries and establishes bounds on information gain, revealing high fidelity reconstructions even with incomplete measurements and connecting the results to random matrix theory.

Contribution

It introduces a novel analysis of quantum tomography with diagonal unitaries, deriving bounds on information gain and linking measurement ensembles to random matrix theory.

Findings

High fidelity reconstruction with diagonal unitaries despite incomplete data

Upper bounds on information gain derived from random matrix ensembles

Eigenvalue distributions of the measurement covariance matrix follow Porter-Thomas distribution

Abstract

We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in a fixed basis even though the measurement record is not informationally complete. We then compare this with the information generated and fidelities obtained by application of a different Haar random unitary at each time step. We give an upper bound on the maximal information that can be obtained in tomography and show that a covariance matrix taken from the Wishart-Laguerre ensemble of random matrices and the associated Marchenko-Pastur distribution saturates this bound. We find that physically, this corresponds to an application of a different Haar random unitary at each time step. We show that repeated application of random diagonal unitaries gives a covariance matrix in tomographic estimation that corresponds to a new ensemble of random matrices. We analytically and numerically estimate eigenvalues of this ensemble and show the information gain to be bounded from below by the Porter-Thomas distribution.