We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy. We find that for small full width at half maximum (FWHM) of the position entropy density, the FWHM of the momentum entropy density is large and vice versa. By increasing the confined potential depth, the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases. By increasing the potential depth, the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases. By increasing the number of wells, the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well. As an example, we might localize the ground state as well as the position entropy densities of the 1st, 2nd, and 6th excited states for a four-well quantum system. Also, we verify the Bialynicki–Birula–Mycieslki (BBM) inequality.
For every
n≥9 that is a multiple of 3, we construct an explicit access structure on
n participants. In every perfect secret-sharing scheme realising this access structure, if
S denotes the random secret, then the sum of the share entropies is at least
(9n2+32n)H(S), and some participant has share entropy at least
(6n+21)H(S). After normalisation by
H(S), these are respectively
Ω(n2) and
Ω(n) lower bounds and also give the same asymptotic lower bounds on the total and largest expected binary lengths of the shares. This improves by a logarithmic factor the longstanding general lower bounds of
Ω(n2/logn) for total share size and
Ω(n/logn) for maximum share size due to Csirmaz. The proof uses only elementary Shannon inequalities, together with some averaging arguments. The Shannon-information method has universal
O(n2) and
O(n) ceilings for the total and maximum normalised entropy lower bounds it can certify, so our construction reaches both ceilings up to constant factors.
FINDING: Von Neumann entropy extends Shannon entropy to quantum systems, with unitarity preserving information in closed systems; topological order in 3D reveals constant entropy signatures. | MATH: Von Neumann entropy
S(ρ)=−Tr(ρlogρ); Shannon entropy
H=−∑pilogpi; unitarity implies
S(ρ(t))=S(ρ(0)) for closed systems; topological entanglement entropy constant
γ=logD where
D is total quantum dimension. | CONNECTION: Topological order entropy constant
γ relates to anyon fusion rules and modular tensor categories, echoing lattice symmetries and root system structures (e.g.,
D2=∑da2 for anyon quantum dimensions
da, reminiscent of Coxeter number relations). No direct golden ratio or base-60 link in these findings. | DEPTH: 7 — Unifies information theory with quantum dynamics and topological phases, but does not introduce new constants or ratios; deepens understanding of information Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com