Complex Quantum Entropy

In (1) (from 2016), a complex quantum entropy is introduced as: S density = - 2 ln(W) W*W, where W(r vector) is the wavefunction. We note that in the single particle bound case for which W*=W, that this entropy is -2 WW ln(W) = - WW ln(WW) which is the well-known Shannon’s result for spatial quantum entropy. (We note that there is usually also a momentum entropy - a(p)a(p) ln(a*(p)a(p)) which is not included in (1), but it is possible that one wishes to focus on spatial entropy alone). In previous notes (2) (from 2021), we introduced a complex quantum entropy - W ln(W) in order to calculate the free particle quantum wavefunction through the MaxEnt procedure. We then maximized this subject to the constraint - ikx W(x) in order to obtain exp(ikx). Here we try to show that the complex entropy of (1) is the same as that of (2), when one links (2) to the single particle bound state spatial entropy -W*W ln(W*W).