Consider you have a complex valued function that is possible to represent in the following form: $$ \Psi(x_1, x_2) = \sum\limits_{j} \lambda_j \psi_j(x_1) \psi_j(x_2) + \sum\limits_k \sigma_k \phi_k(|x_1-x_2|) \xi_k(x_<) , $$ where $x_<$ is the smallest of $x_1, x_2$. As this is complex-valued, only $|\Psi(x_1, x_2)|^2$ is an actual probability distribution with $\int \int |\Psi(x_1, x_2)|^2 dx_1 dx_2= 1$, $\lambda_j, \sigma_k$ are real numbers (but the functions $\phi_j, \phi_k, \xi_k$ can be complex valued in general).
My question is the following: how to introduce a continuous measure that gives $0$ if $\Psi(x_1, x_2)$ contains only terms of the first sum, where the variables are independent, and $1$ if it contains only the "correlated" terms? Please, note, that it doesn't seem to be the covariance for the obvious reasons.
After some googling, I think, I have finally found a proper measure, which is called https://en.wikipedia.org/wiki/Distance_correlation
This measure is just better than a regular covariance as it is always positive, and between $0$ (uncorrelated variables), and $1$ (strongly correlated variables).