Entropy of pure states: not all wave functions are born equal

Denis Sunko
Department of Physics
Faculty of Science, University of Zagreb
Zagreb, Croatia

Many-fermion Hilbert space is a finitely generated free module [1]. The generators of the module, called shapes, are precisely all N-fermion primitive realizations of the Pauli principle. They are geometric objects in wave-function space, which constrain fermion motion kinematically. Although the number of shapes is absolutely very large, N!d-1 for N fermions in d dimensions, it is finite and vanishingly small relative to the total number of excited states in the same energy range. The majority of states, called trivial, can be constructed as bosonic excitations of the shapes. This algebraic structure is analysed from the point of view of entropy. The shapes naturally form a lattice partially ordered by information content, because all can be obtained as symmetrized derivatives of a product of Vandermonde forms [2]. However, because they are all pure states, their von Neumann entropy and all related so-called quantum entropies are zero. It is shown that logical entropy [3] correctly encodes the information content of the shapes, when each difference appearing in the defining Vandermonde forms is interpreted as a distinction. With this interpretation, logical entropy becomes the only quantum entropy sensitive to the fundamental free-module structure of Hilbert space.

[1] D. K. Sunko, Natural generalization of the ground-state Slater determinant to more than one dimension. Physical Review A 93, 062109 (2016) DOI: 10.1103/PhysRevA.93.062109

[2] D. K. Sunko, Many-Fermion Wave Functions: Structure and Examples. In: Bonča J., Kruchinin S. (eds) Advanced Nanomaterials for Detection of CBRN. NATO Science for Peace and Security Series A: Chemistry and Biology. Springer, Dordrecht (2020). DOI: 10.1007/978-94-024-2030-2_5

[3] D. Ellerman, Logical information theory: new logical foundations for information theory, Logic Journal of the IGPL, Volume 25, Issue 5, October 2017, Pages 806–835, DOI: 10.1093/jigpal/jzx022