Statistical analysis of sampling methods in quantum tomography

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 noncommuting 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.

Figure 1

Scheme for overlapping a signal state ${\Phi }_0\left(\beta \right)$ with a coherent state.

Figure 2

Nonclassicality quasiprobability of a squeezed state with $V_x=0.5$ and $V_p=2.0$.

Figure 3

Standard deviation from balanced homodyne measurements of the nonclassicality quasiprobability in Fig. 2.

Figure 4

Standard deviation from unbalanced homodyne measurements of the nonclassicality quasiprobability in Fig. 2.

Figure 5

Comparison of the results for balanced and unbalanced homodyne tomography. The blue shaded area corresponds to one standard deviation.