A Note on Quantum Entropy

Noboru Watanabe

Makoto Ohya, D�nes Petz

Francesco R. Ruggeri

In a previous note (1) we suggested the existence of a quantum entropy associated with a free quantum particle with momentum p and wavefunction exp(ipx) due to the presence of a wavelength. The usual approach of using P(x)=W*(x)W(x) in Shannon’s entropy does not capture this entropy as it yields a constant for all p values. Thus we suggested using |cos(px)| C where C is a normalization constant over half a wavelength. This value may be used in Shannon’s entropy formula integrated over a half wavelength to give an entropy per crest. We then extended the idea to a quantum bound state. In this note we wish to examine the relationship of this entropy to momentum i.e. energy flow. In (1) we noted that this entropy does not satisfy the first law of thermodynamics. Here we argue that it also does not satisfy the second law using ideas from refraction. Finally we note that quantum thermodynamical calculations using temperature seem to ignore this entropy. In fact, these calculations ignore the usual P(x)=W*W or P(p) = a*(p)a(p) Shannon’s entropies for a pure state suggesting that only some “uncertainty” is included in typical entropy calculations while other uncertainty may apparently be ignored.

Francesco R. Ruggeri

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).

Dominik Janzing

Francesco R. Ruggeri

Recently, there has been some interest in the literature in calculating temperature dependent quantum mechanical wavefunctions using the principle of maximum entropy. The idea seems to be to obtain a differential equation, similar to the Schrodinger equation, but with temperature dependent terms. It should be noted that this does not follow von Neumann statistical quantum mechanics. It should also be noted that other authors in the literature maximize or minimize other quantities related to information theory to calculate a wavefunction. The objective of this note is not to examine the maximization of entropy as applied to the calculation of a wavefunction, but to suggest a different form for entropy. Traditionally, authors seem to use the Shannon form -k W*(x)W(x) ln[ W*(x)W*(x)] as a spatial entropy and exchange W(x) for fp for the momentum one, with W(x) being the wavefunction. It is the spatial form which they seem to maximize in calculations. In this note, we consider a probability P(x intersection x) = W(x) fp sin(px) and use it in Shannon’’s entropy. Maximization of entropy then leads to an integral equation which, it seems, needs to be solved numerically, We can however, multiply such an equation by fp and sum over p to obtain results which can be compared with those of authors who use -k W*(x)W(x) ln[ W*(x)W*(x)] . NOTE: In later preprints, we have used W(x) fp exp(ipx) as probability and calculated Shannon's entropy. This has given an overall result for entropy including a term for Shannon's entropy for W(x), another term with Shannon's entropy for fp and a final term W(x) x d/dx W(x). This last term, linked to exp(ipx), is also equal to the covariance of x and d/dx ln(W(x)) which is used in the derivation of the Cramer-Rao inequality for Fisher information.