Two measurements of $A$ and $B$ are carried out one after the other. The measurements of $A$ are controlled by the parameter $λ_A$ in the Kraus operator, where the measurements of $B$ are controlled by the parameter $λ_B$. 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 $λ_A$ and $λ_B$. 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 $λ_A$ and $λ_B$. 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 $λ_A$ and not on the value of $λ_B$. 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.
Predicting properties of quantum thermal states from a single trajectory
Simons Institute for the Theory of Computing, University of California, Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Department of Mathematics, University of California, Berkeley（加州大学伯克利分校数学系）；Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室应用数学与计算研究部）
