Partial order and topology of Hermitian matrices and quantum Choquet integrals for density matrices with given expectation values

TL;DR

This paper explores the mathematical structure of Hermitian matrices as a partially ordered and topological space, and applies these concepts to develop quantum Choquet integrals for density matrices with specified expectation values.

Contribution

It introduces a topological and order-theoretic framework for Hermitian matrices and applies it to quantum Choquet integrals, enabling new methods for density matrix reconstruction.

Findings

Hermitian matrices form a $T_0$ topological space.

Density matrices form a Hausdorff totally disconnected space.

Application to reconstruct density matrices with given expectation values.

Abstract

The set $M$ of $d\times d$ Hermitian matrices (observables) is studied as a partially ordered set with the L\"{o}wner partial order. Upper and lower sets in it, define the concept of cumulativeness (used mainly with scalar quantities) in the context of Hermitian matrices. Partial order and topology are intimately related to each other and the set $M$ of Hermitian matrices is also studied as a topological space, where open and closed sets are the upper and lower sets. It is shown that the set $M$ of Hermitian matrices is a $T_0$ topological space, and its subset ${\mathfrak D}$ of density matrices is Hausdorff totally disconnected topological space. These ideas are a prerequisite for studying quantum Choquet integrals with Hermitian matrices (as opposed to classical Choquet integrals with scalar quantities). Capacities (non-additive probabilities), cumulative quantities that involve Hermitian matrices, and M\"obius transforms that remove the overlaps between non-commuting observables, are used in quantum Choquet integrals. An application of the formalism is to find a density matrix, with given expectation values with respect to $n$ (non-commuting) observables. Examples of calculations of such a density matrix (with quantified errors in its expectation values), are presented.