Bound State Entropy as Mutual Information?

   In a previous note (1) we suggested using P(x)P(p/x) where P(x)=W(x)W(x) and P(p/x) = a(p)exp(ipx)/W(x) for a bound state (where W(x) is the wavefunction) in Shannon’s entropy expression - Sum over i P(i) ln(P(i)). This yields a bound state entropy of S= .5Sp+ .5Sx.

Historically (2) Sp+Sx, called entropic uncertainty, has been of interest as an alternative to the Heisenberg uncertainty relationships for p and x. 

   In this note we argue that this sum of momentum and spatial entropies is directly linked to the idea of mutual information from information theory. In (3) it is noted that mutual information is “the amount of information that can be obtained about one random variable by observing another”. Thus this seems to be directly related to uncertainty in two p and x.