Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Simple Theorem on Hermitian Matrices and an Application to the Polarization of Vector Particles

Daniel Zwanziger

Published Oct 26, 1964• 6 citations
Publisher

Abstract

It is proven that a necessary and sufficient condition for an $n$-dimensional Hermitian matrix $\ensuremath{\rho}$ to be positive definite is that it be expressible in the form $\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}$, where $O$ is a complex orthogonal matrix and $E$ is a diagonal matrix with positive elements. This accomplishes a parametrization since $O$ has ${n}^{2}\ensuremath{-}n$ real parameters and $E$ has $n$ of them. The proof is constructive, giving $O$ and $E$. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.

DOI
10.1103/physrev.136.b558

Related papers

REMARKS ON BELL'S INEQUALITY FOR SPIN CORRELATIONS

K. Balasubramanian, J. C. Gupta, Kalyanapuram Rangachari Parthasarathy

where ei,^2,..., en are ?1. It is not difficult to construct real non-negative definite matrices E = ((crij)), 1 5! hj 5: n, satisfying an = 1 for every i but not (1.1). Thus there arises the natural problem of finding simple verifiable conditions on E in order to ensure that it is the correlation matrix of n spin variables. Here we shall find such necessary and sufficient conditions when n = 3 or 4. We also obtain a characterisation of all correlation matrices of n exchangeable spin variables. By using the observables of a fermion field obeying the canonical anticom mutation relations it is possible to realize any non-negative matrix with diagonal entries unity as the correlation matrix of spin observables.

Sankhya. Series A•Jan 1, 1998•7 citations

Real eigenvalues of non-hermitian operators

Péter R. Śurján, Ágnes Szabados, András Gombás

A basic fact, having fundamental significance in quantum mechanics, is that hermitian (or self-adjoint) operators have only real eigenvalues. However, in certain applications in molecular physics, one deals with non-hermitian operators. We discuss a condition for non-hermitian operators to have real eigenvalues, proving that it is the case if and only if it can be decomposed as a product of two, generally non-commuting hermitian operators, one of which is positive definite. The theorem is illustrated on the example of non-hermitian effective Hamiltonians occurring in the non-perturbative form of the Bloch equation.

Molecular Physics•Nov 24, 2023•1 citation

Degeneracy of the eigenvalues of hermitian matrices

W.J. Caspers

Degeneracy of the eigenvalues of hermitian matrices is analyzed in terms of algebraic relations between the matrix elements. Only finite-dimensional matrices are considered, representing Hamiltonians of spin systems in particular.The analysis is in terms of linear relations in the space of 'diagonal' operators, i.e. a complete set of powers of the Hamiltonian. A relation is found with the theorem formulated by Von Neumann and Wigner that only the variation of three parameters of the Hamiltonian may result in a 'crossing' or degeneracy of two eigenvalues.

Journal of Physics Conference Series•Mar 1, 2008•4 citations

Causality and the Complete Positivity of Classical Polarization Maps

Omar Gamel, Daniel F. V. James

The most general physical transformation on the polarization state has been represented as an ensemble of Jones matrices, as ∑ i V i Φ V i †. But this has been either directly assumed or proven through non-rigourous means. We derive this expression from simple physical principles and the matrix theory of positive maps.

Frontiers in Optics•Jan 1, 2011•1 citation

Parametrization of Quantum States and Channels

T. Constantinescu, Viswanath Ramakrishna

In this manuscript, a parametrization of positive matrices together with some of its many applications in quantum information theory is given.

arXiv:quant-ph/0306167•Jun 25, 2003•0 citations

Diagonal N-representability as a method for solving the representability problem

M. M. Mestechkin

The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).

Russian Journal of Physical Chemistry B•Mar 1, 2017•0 citations
It is proven that a necessary and sufficient condition for an nnn-dimensional Hermitian matrix \ensuremathρ\ensuremath{\rho}\ensuremathρ to be positive definite is that it be expressible in the form \ensuremathρ=OEO\ifmmode†\else\textdagger\fi\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}\ensuremathρ=OEO\ifmmode†\else\textdagger\fi, where OOO is a complex orthogonal matrix and EEE is a diagonal matrix with positive elements. This accomplishes a parametrization since OOO has n2\ensuremath−n{n}^{2}\ensuremath{-}nn2\ensuremath−n real parameters and EEE has nnn of them. The proof is constructive, giving OOO and EEE. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.
where ei,^2,..., en are ?1. It is not difficult to construct real non-negative definite matrices E = ((crij)), 1 5! hj 5: n, satisfying an = 1 for every i but not (1.1). Thus there arises the natural problem of finding simple verifiable conditions on E in order to ensure that it is the correlation matrix of n spin variables. Here we shall find such necessary and sufficient conditions when n = 3 or 4. We also obtain a characterisation of all correlation matrices of n exchangeable spin variables. By using the observables of a fermion field obeying the canonical anticom mutation relations it is possible to realize any non-negative matrix with diagonal entries unity as the correlation matrix of spin observables.
A basic fact, having fundamental significance in quantum mechanics, is that hermitian (or self-adjoint) operators have only real eigenvalues. However, in certain applications in molecular physics, one deals with non-hermitian operators. We discuss a condition for non-hermitian operators to have real eigenvalues, proving that it is the case if and only if it can be decomposed as a product of two, generally non-commuting hermitian operators, one of which is positive definite. The theorem is illustrated on the example of non-hermitian effective Hamiltonians occurring in the non-perturbative form of the Bloch equation.
Degeneracy of the eigenvalues of hermitian matrices is analyzed in terms of algebraic relations between the matrix elements. Only finite-dimensional matrices are considered, representing Hamiltonians of spin systems in particular.The analysis is in terms of linear relations in the space of 'diagonal' operators, i.e. a complete set of powers of the Hamiltonian. A relation is found with the theorem formulated by Von Neumann and Wigner that only the variation of three parameters of the Hamiltonian may result in a 'crossing' or degeneracy of two eigenvalues.
The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).