Quantum correlation effects on two successive measurements that are presented by non-commuting operators

Abstract

Two measurements of $A$ and $B$ are carried out one after the other. The measurements of $A$ are controlled by the parameter $\lambda_A$ in the Kraus operator, where the measurements of $B$ are controlled by the parameter $\lambda_B$. Strong measurements imply that the parameters in the Kraus operators approach infinite large values while weak measurements are carried out when the parameters approach zero. Here we prove that by repeating on the two successive measurements of $A$ and $B$ then: (1) Average over all measurements of $A$ is invariant of the measurement strength parameters $\lambda_A$ and $\lambda_B$. It implies that all surprising results obtained in weak measurements of $A$ are washed out when the average is taken. (2) If the operators $\hat A$ and $\hat B$ commute then the mean value of $B$ as obtained by taking the average of the results for $B$ over all measurements is invariant of $\lambda_A$ and $\lambda_B$. Moreover it is exactly equal to the expectation value of $\hat B$ as expected for strong measurements of $B$. (3) If $\hat A$ and $\hat B$ do not commute \textit{and} another condition given in this paper is satisfied then the mean value of the results obtained for $B$ depends on the value of $\lambda_A$ and not on the value of $\lambda_B$. An illustrative possible experiment to show the effect of the strength of the measurements of A on the results obtained for the measurements of B is given.