A comparative study of estimation methods in quantum tomography

Abstract

As quantum tomography is becoming a key component of the quantum engineering toolbox, there is a need for a deeper understanding of the multitude of estimation methods available. Here we investigate and compare several such methods: maximum likelihood, least squares, generalised least squares, positive least squares, thresholded least squares and projected least squares. The common thread of the analysis is that each estimator projects the measurement data onto a parameter space with respect to a specific metric, thus allowing us to study the relationships between different estimators.The asymptotic behaviour of the least squares and the projected least squares estimators is studied in detail for the case of the covariant measurement and a family of states of varying ranks. This gives insight into the rank-dependent risk reduction for the projected estimator, and uncovers an interesting non-monotonic behaviour of the Bures risk. These asymptotic results complement recent non-asymptotic concentration bounds of [36] which point to strong optimality properties, and high computational efficiency of the projected linear estimators.To illustrate the theoretical methods we present results of an extensive simulation study. An app running the different estimators has been made available online.

Reconstructing Quantum States With Quantum Reservoir Networks

“…In such situations, a maximum likelihood estimation can be employed to find out the closest physical density matrix. A wealth of study has been devoted to explore this estimation [40]- [44]. When needed, these methods can directly be applied as a final step in the QRST scheme.…”

On quantum tomography on locally compact groups

“…ρ ∈ S(H), where unitary operators π l (g) are determined by (8). It follows from Lemma 4 that Z n × Z n = ⊕ n l=0 G l .…”

“…Also, the measurements themselves are made not by a continuum, but by a finite number. The problem of the accuracy of the reconstruction of a quantum state from the partial information about its tomogram is extremely relevant [8]. Earlier the Wigner function was introduced on Lie groups [9] but no tomography (sets of probability distributions determining a state) was defined.…”

Quantum Tomography of Pure States with Projective Measurements Distorted by Experimental Noise

“…In order to estimate the parameters t 1 , t 2 , t 3 , t 4 , we shall apply the method of least squares (LS) [24]. This method has been used to study the performance of QST frameworks with simulated measurement results [25,26]. According to the LS method, the minimum value of the following function needs to be determined:…”