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

Denis K. Sunko,Giovanni Manfredi

doi:10.1051/fopen/2021006

Abstract

Many-body Hilbert space has the algebraic structure of a finitely generated free module. All N-body wave functions in d dimensions can be generated by a finite number of N!d − 1 of generators called shapes, with symmetric-function coefficients. Physically the shapes are vacuum states, while the symmetric coefficients are bosonic excitations of these vacua. It is shown here that logical entropy can be used to distinguish fermion shapes by information content, although they are pure states whose usual quantum entropies are zero. The construction is based on the known algebraic structure of fermion shapes. It is presented for the case of N fermions in three dimensions. The background of this result is presented as an introductory review.

Highlights

The realization that the space of physical states is a Hilbert space is arguably the milestone separating Bohr’s “old”quantum mechanics from its modern form
In the language of logical entropy, the number of terms in the Vandermonde product counts the number of distinct pairs of variables
We are really looking for distinctions in excess of the minimum required by the Pauli principle, which is that a wave function changes sign when triplets and are exchanged

Summary

Introduction

The realization that the space of physical states is a Hilbert space is arguably the milestone separating Bohr’s “old”quantum mechanics from its modern form. The algorithm is just taking derivatives of polynomials, so it reduces the information content of shape wave functions in each step. The algorithm finds generators of three-dimensional many-fermion Hilbert space over the space of bosonic excitations as symmetrized derivatives of the triple product (source shape), DN 1⁄4 ÁN ðtÞÁN ðuÞÁN ðvÞ; ð7Þ

Results

Conclusion