Physical Review Letters

Orthogonalization Process by Recurrence Relations

M. Howard Lee

Published Oct 11, 1982• 166 citations

Abstract

An orthogonalization process is proposed, applicable to spaces which are realizations of abstract Hilbert space. It is simpler than the Gram-Schmidt process. A recurrence relation which orthogonalizes a physical space is proposed and it is shown that the generalized Langevin equation is contained therein. This process serves as a basis for understanding the nature of the dynamic many-body formalism.

DOI: 10.1103/physrevlett.49.1072

Related papers

Method of Recurrence Relations and Applications to Many-Body Systems

M. Howard Lee, J. M. Hong, J. Florencio

The method of recurrence relations allows one to calculate time dependent correlation functions from first principles. It originates from the Kubo scalar product which realizes abstract Hilbert space of dynamical variables. For this realized space, we show that there exists a unique orthogonalization process by recurrence relation. The method of recurrence relations exploits geometric properties of the structure of the realized Hilbert space such as the dimensionality and shape. These geometric properties depend on models as well as static constraints, e.g., temperature, wave vector, size. This method has been applied to several physical models including the homogeneous dense electron gas, the spin-1/2 XY and transverse Ising models, the spin van der Waals model and others. For these models, we have obtained the relaxation function, response function and memory function.

Physica ScriptaJan 1, 198757 citations

On orthogonal bases in the Hilbert-Schmidt space of matrices

Jens Siewert

Abstract Decomposition of (finite-dimensional) operators in terms of orthogonal bases of matrices has been a standard method in quantum physics for decades. In recent years, it has become increasingly popular because of various methodologies applied in quantum information, such as the graph state formalism and the theory of quantum error correcting codes, but also due to the intensified research on the Bloch representation of quantum states. In this contribution we collect various interesting facts and identities that hold for finite-dimensional orthogonal matrix bases.

arXiv:2205.06035Journal of Physics CommunicationsMay 1, 202215 citations

Fermionic quantum orthogonalizations I

Gyula Lakos

We generalize classical orthogonalization procedures from real linear algebra to the setting of fermionic quantum (FQ) operations. In the case of the Gram-Schmidt orthogonalization procedure, the generalization is easy. This, however, helps to obtain general information regarding FQ operations, and to generalize the symmetric orthogonalization procedure.

arXiv:1509.08675Sep 29, 20152 citations

UNITARY OPERATORS, ENTANGLEMENT, AND GRAM–SCHMIDT ORTHOGONALIZATION

Yorick Hardy, Willi-Hans Steeb

We consider finite-dimensional Hilbert spaces [Formula: see text] with [Formula: see text] with n ≥ 2 and unitary operators. In particular, we consider the case n = 2m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces [Formula: see text] are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.

International Journal of Modern Physics CJun 1, 20090 citations

On the orthogonalization of quantum states

Yong Xiao, Guang Yang

Some quantum states can dynamically evolve to an orthogonal state after a time interval, while some states can never evolve to their orthogonal states. In this paper, we investigate three issues on the orthogonalization properties of quantum states. First, we answer the question as to under what conditions an initial state can evolve to an orthogonal state. Second, we discuss the orthogonalization rate of quantum states, i.e. , the total number of orthogonal states that the evolution can go through in unit time. Finally, we show that the Margolus-Levitin bound for orthogonalization time remains tight for superpositions of more than two eigenstates.

Europhysics Letters (EPL)Jul 1, 20131 citation

Statistical Basis of Physical Laws

R. Collins

The premise that physical variables must be viewed as random variables, with observed values, assigned as averages over distributions, is shown to yield the formalisms of thermodynamics, Hamiltonian mechanics, and quantum theory. A general mathematical argument first establishes the validity of an embedding of th space of physical variables in a larger manifold of "hidden" variables and a general structure for the distribution of these variables. Then, by consideration of various ways of specifying the preparation of a system for measurement, the formalisms of physics cited above are deduced. It is also shown that the generalized Langevin equation follows from these premises and is applicable to a wide class of phenomena. This formulation yields a new definition for compatible observables as used in quantum theory and demonstrates, in a natural way, that physical laws should be invariant under orthogonal transformations of certain variables, which in one circumstance include the complex orthogonal transformations of the Lorentz form, as in special relativity. In summary, it appears that most of the formal structure of physical theory is a consequence of the statistical nature of physical variables.

Physical ReviewJul 25, 19695 citations