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arXiv:2506.06794 · Journal of Geometry and Physics

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

Apostol Vourdas

Published Jun 3, 2025• 1 citation

Abstract

The set M of d × d Hermitian matrices (observables) is studied as a partially ordered set with the Löwner 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 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 Möbius 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.

DOI: 10.1016/j.geomphys.2025.105555

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Abstract In a quantum system with d -dimensional Hilbert space, the Q -function of a Hermitian positive semidefinite operator θ , is defined in terms of the d 2 coherent states in this system. The Choquet integral of the Q -function of θ , is introduced using a ranking of the values of the Q -function, and Möbius transforms which remove the overlaps between coherent states. It is a figure of merit of the quantum properties of Hermitian operators, and it provides upper and lower bounds to various physical quantities in terms of the Q -function. Comonotonicity is an important concept in the formalism, which is used to formalize the vague concept of physically similar operators. Comonotonic operators are shown to be bounded, with respect to an order based on Choquet integrals. Applications of the formalism to the study of the ground state of a physical system, are discussed. Bounds for partition functions, are also derived.

Density matrices defined with respect to a finite basis set are considered as elements in a vector space. A basis set is introduced in the space whose elements are Hermitian matrices. By a suitable transformation of the basis, one component only will contribute to the trace, so the space of unit trace matrices is a translated linear subspace. The positivity constraint is then examined in terms of distance from a multiple of the unit matrix, which is in the interior of the set of density matrices. The boundary of this set is shown to lie between two concentric hyperspheres. The set whose elements are

N

-representable 1 matrices has similar properties.

The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of

p

particles for each value of

p

. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the

N

-representability problem is given, and two sufficient conditions and one necessary condition for

N

representability are given in terms of distances and norms.

In this paper we review an approach to the dynamics of open quantum systems based of non-Hermitian Hamiltonians. Non-Hermitian Hamiltonians arise naturally when one wish to study a subsystem interacting with a continuum of states. Moreover, quantum subsystems with probability sinks or sources are naturally described by non-Hermitian Hamiltonians. Herein, we discuss a non-Hermitian formalism based on the density matrix. We show both how to derive the equations of motion of the density matrix and how to define statistical averages properly. It turns out that the laws of evolution of the normalized density matrix are intrinsically non-linear. We also show how to define correlation functions and a non-Hermitian entropy with a non zero production rate. The formalism has been generalized to the case of hybrid quantum-classical systems using a partial Wigner representation. The equations of motion and the statistical averages are defined analogously to the pure quantum case. However, the definition of the entropy requires to introduce a non-Hermitian linear entropy functional.

Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.