Statistical analysis of sampling methods in quantum tomography

TL;DR

This paper analyzes the statistical uncertainties in quantum tomography sampling methods, showing when estimates reach the fundamental quantum level of uncertainty and comparing different measurement approaches.

Contribution

It provides a detailed comparison of sampling methods in quantum tomography, highlighting conditions for achieving quantum-level uncertainty and evaluating measurement techniques.

Findings

Sampling estimates match quantum uncertainty when based on a single measurable operator.

Non-commuting observables lead to higher statistical uncertainties.

Balanced homodyne tomography does not reach quantum uncertainty, unlike unbalanced detection.

Abstract

In quantum physics, all measured observables are subject to statistical uncertainties, which arise from the quantum nature as well as the experimental technique. We consider the statistical uncertainty of the so-called sampling method, in which one estimates the expectation value of a given observable by empirical means of suitable pattern functions. We show that if the observable can be written as a function of a single directly measurable operator, the variance of the estimate from the sampling method equals to the quantum mechanical one. In this sense, we say that the estimate is on the quantum mechanical level of uncertainty. In contrast, if the observable depends on non-commuting operators, e.g. different quadratures, the quantum mechanical level of uncertainty is not achieved. The impact of the results on quantum tomography is discussed, and different approaches to quantum tomographic measurements are compared. It is shown explicitly for the estimation of quasiprobabilities of a quantum state, that balanced homodyne tomography does not operate on the quantum mechanical level of uncertainty, while the unbalanced homodyne detection does.