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Journal of Physics A Mathematical and Theoretical

On the single-exponential closed form of the product of two exponential operators

Tiberiu S. Tudor

Published Nov 21, 2007• 3 citations
Publisher

Abstract

In the frame of a vectorial Pauli algebraic approach it is shown that the product of two exponentials of any two-by-two linear operators can be put in a single-exponential closed form. As a first application, a compact vectorial expression for the characteristics (angle and axis) of the product of two R 3 rotations is established. The same mathematics can be used in a large diversity of problems of the whole class of two-states physical systems. An exemplification in the field of polarization optics is given.

DOI
10.1088/1751-8113/40/49/013

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In the frame of a vectorial Pauli algebraic approach it is shown that the product of two exponentials of any two-by-two linear operators can be put in a single-exponential closed form. As a first application, a compact vectorial expression for the characteristics (angle and axis) of the product of two R 3 rotations is established. The same mathematics can be used in a large diversity of problems of the whole class of two-states physical systems. An exemplification in the field of polarization optics is given.
Abstract Some problems in elementary geometry are approached from the point of view of linear algebra and generalized to the theory of linear spaces of finite or infinite dimensions having a positive definite binary product. The angle ω between two elements of the linear space is defined from the concept of length by means of the cosine‐theorem. A rotation is then defined as a special case of a unitary transformation moving all elements the same angle ω, except that under certain circumstances, some elements may stay invariant. In the former case, one speaks of a rotation around an “external axis,” and in the latter case, of a rotation around an “internal axis” defined by the invariant elements. It is shown that the finite rotations U of both types may be expressed in the simple exponential form U = exp(iωm), where the “generator” m in the former case is an operator satisfying the relation m2 = 1, and in the latter case, m3 = m. The structure of the group of finite rotations in the former case is clarified in some detail. As an illustration of the theory, some applications to the three‐ and two‐dimensional spaces as well as to the theory of spin are given. The coupling between the ordinary three‐dimensional rotations and the spinor transformations is considered in somewhat greater detail.
A unified treatment of the Pauli algebraic forms of the linear operators defined on a unitary linear space of two dimensions over the field of complex numbers C(1) is given. The Pauli expansions of the normal and nonnormal operators, unitary and Hermitian operators, orthogonal projectors, and symmetries are deduced in this frame. A geometrical interpretation of these Pauli algebraical results is given. With each operator, one can associate a generally complex vector, its Pauli axis. This is a natural generalization of the well-known Poincaré axis of some normal operators. A geometric criterion of distinction between the normal and nonnormal operators by means of this vector is established. The results are exemplified by the Pauli representations of the normal and nonnormal operators corresponding to some widespread composite polarization devices.
It is well-known that any two pure quantum states (in the same Hilbert space) can be mapped to any other using unitary transformations. However, previous approaches to this problem required two explicit bases for the Hilbert space, one each for the initial and target states, and thus their complexity necessarily scales with the dimension of the Hilbert space. In this Letter, we show how to utilize novel algebraic methods to construct a closed-form exponential unitary transformation which achieves this in general, using only a single unitary generator. This construction is independent of any bases and agnostic to the dimension of the Hilbert space. We highlight the usefulness of this tool for studying relationships between systems of pure states in quantum information theory, as well in elementary analyses of quantum circuits and unitary operators.
This chapter provides a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. It also provides some useful mathematical tools to derive important theoretical results and expressions in a simple and elegant way. In quantum mechanics the state of a system is represented by an element of a particular vector space, in which one defines an appropriate scalar or inner product, known in mathematics as Hilbert space. Hermitian operators with a complete set of eigenvectors belonging to that state space represent physical observables. Moreover, in certain cases it proves useful to work with vector spaces formed by linear operators. Hermitian operators are particular cases of linear operators and physical observables are particular cases of Hermitian operators. In many physical applications of operator methods one needs to reorder products of exponential operators or to rewrite one such operator as a product of simpler ones.
The problem of the time-varying polarization devices is analyzed in the general frame of the dynamics of the two-state systems. A vectorial and pure operatorial Pauli algebraic approach, conjugated with a Poincaré geometric one is adopted. The general equation of evolution of the density operator of the light polarization state interacting with a time-varying birefringent device is established this way and applied in analyzing the modification of the polarization and spectral structure of light by modulation.
We present a basis-independent, purely operator, derivation of the expression for the finite-angle rotation operator about a fixed axis. The standard matrix representation of the operator, as well as the standard vectorial expression for the related finite-angle rotation formula, are easily recovered. R´ esum´ e : Nous obtenons l'operateur de rotation pour un angle fini autour d'un axe fixe par une methode independante de la base en n'utilisant que des operateurs. Nous retrouvons facilement la repesentation matricielle standard de l'operateur ainsi que l'expression vectorielle usuelle pour la rotation d'un angle fini. (Traduit par la r ´