Equivalence classes of coherent projectors in a Hilbert space with prime dimension: Q functions and their Gini index

TL;DR

This paper introduces a classification of coherent subspaces in a prime-dimensional Hilbert space, explores their properties, and uses economic inequality measures to analyze quantum phase space distributions.

Contribution

It defines equivalence classes of coherent projectors, studies their properties, and introduces Gini index-based measures for quantum state distributions.

Findings

Coherent subspaces form equivalence classes with closure properties.

Different classes provide various granularizations of the Hilbert space.

Gini index quantifies inequality in quantum phase space distributions.

Abstract

Coherent subspaces spanned by a finite number of coherent states are introduced, in a quantum system with Hilbert space that has odd prime dimension $d$. The set of all coherent subspaces is partitioned into equivalence classes, with $d^2$ subspaces in each class.The corresponding coherent projectors within an equivalence class, have the `closure under displacements property' and also resolve the identity. Different equivalence classes provide different granularisation of the Hilbert space, and they form a partial order `coarser' (and `finer'). In the case of a two-dimensional coherent subspace spanned by two coherent states, the corresponding projector (of rank $2$) is different than the sum of the two projectors to the subspaces related to each of the two coherent states. We quantify this with `non-addditivity operators' which are a measure of quantum interference in phase space, and also of the non-commutativity of the projectors. Generalized $Q$ and $P$ functions of density matrices, which are based on coherent projectors in a given equivalence class, are introduced. Analogues of the Lorenz values and the Gini index (which are popular quantities in Mathematical Economics) are used here to quantify the inequality in the distribution of the $Q$ function of a quantum state, within the granular structure of the Hilbert space....