Weak covariance and the correlation of an observable with pre-selected and post-selected state energies during its time-dependent weak value measurement

Abstract

The peculiar weak energy of evolution appears as a factor in the equation of motion $$ \dot{A}_{w}$$ for a time-dependent weak value of an observable $$\hat{A}$$ . This energy has the mathematical form of the weak value of the difference between the two Hamiltonian operators $$\hat{H}_i $$ and $$\hat{H}_f $$ that describe the evolution of the associated pre- and post-selected states, respectively. Here, the weak covariance $$\mathrm{cov}_w \left( {\hat{X},\hat{Y}} \right) $$ for operators $$\hat{X}$$ and $$\hat{Y}$$ is introduced and it is shown that $$ \left| {\dot{A}_{w}} \right| $$ can be expressed entirely in terms of $$\mathrm{cov}_w \left( {\hat{H}_f ,\hat{A}} \right) $$ , $$\mathrm{cov}_w \left( {\hat{A},\hat{H}_i } \right) $$ , and an angle $$\theta $$ that is governed by the complex valued nature of the terms defining each covariance. Several cases are briefly discussed and an experiment is used to illustrate the $$\hat{H}_i =\hat{0}\ne \hat{H}_f $$ case. It is shown that $$cov_w \left( {\hat{H}_f ,\hat{A}} \right) $$ is observed in the associated experimental data.