Inversion of moments to retrieve joint probabilities in quantum sequential measurements

A sequence of moments obtained from statistical trials encodes a classical probability distribution. However, it is well known that an incompatible set of moments arises in the quantum scenario, when correlation outcomes associated with measurements on spatially separated entangled states are considered. This feature, viz., the incompatibility of moments with a joint probability distribution, is reflected in the violation of Bell inequalities. Here, we focus on sequential measurements on a single quantum system and investigate if moments and joint probabilities are compatible with each other. By considering sequential measurement of a dichotomic dynamical observable at three different time intervals, we explicitly demonstrate that the moments and the probabilities are inconsistent with each other. Experimental results using a nuclear magnetic resonance system are reported here to corroborate these theoretical observations, viz., the incompatibility of the three-time joint probabilities with those extracted from the moment sequence when sequential measurements on a single-qubit system are considered.

Figure 1

Circuit for finding three-time probability. Grouped gates represent measurement in the rotated basis, and controlled gates can be cnot or anti-cnot, as explained in the text.

Figure 2

(a) The molecular structure of trifluoroiodoethylene, (b) the corresponding chemical shifts and $J$-coupling values (in Hz), and (c) the pulse sequence for the preparation of initial state. The open pulses are $\pi$ pulses and $\tau =1/\left(4J_{23}\right)$.

Figure 3

Moussa protocol for obtaining the three-time correlated moments. One- and two-time moments can be calculated using the appropriate number of controlled gates.

Figure 4

Moments obtained experimentally from the Moussa protocol. The symbols represent experimentally obtained values of the indicated moments, with the solid lines showing the corresponding theoretical values.

Figure 5

Three-time joint probabilities. The solid curve represents the probabilities obtained directly, and the dashed curve shows the probabilities obtained by inverting the moments. The symbols represent the experimental data.