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Dirac Contour Representation for Quantum Systems with Finite-Dimensional Hilbert Space in the Extended Complex Plane

Aisha Faraj Abukhzam Mohammed, Ismail Mageed

Published Mar 27, 2024• 0 citations
Publisher

Abstract

The Dirac contour representation functions fk(z) and fb(z) are employed to represent theket states |f ⟩ and bra states ⟨f |, respectively, in quantum systems with a finite-dimensional Hilbertspace H_2j+1. The scalar product within these quantum systems is defined using a contour integral.Moreover, a numerical approach is utilized to examine the time evolution of both periodic and non-periodic systems, utilizing several Hamiltonian matrices. Furthermore, the stability of periodic systemsis investigated. In addition to these aspects, we study the most significant application of the Dirac con-tour representation, which is its capability to handle an extended Hilbert space, suitable for describingquantum physics at negative temperatures.

DOI
10.4028/p-kbygq1

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The Dirac contour representation functions fk(z) and fb(z) are employed to represent theket states |f ⟩ and bra states ⟨f |, respectively, in quantum systems with a finite-dimensional Hilbertspace H_2j+1. The scalar product within these quantum systems is defined using a contour integral.Moreover, a numerical approach is utilized to examine the time evolution of both periodic and non-periodic systems, utilizing several Hamiltonian matrices. Furthermore, the stability of periodic systemsis investigated. In addition to these aspects, we study the most significant application of the Dirac con-tour representation, which is its capability to handle an extended Hilbert space, suitable for describingquantum physics at negative temperatures.
This research explores analytic and Dirac contour representation systems with finite d-dimensional Hilbert space, where the dual variables, position x, and momentum p, take values in Z(d)(the integer modulo d). It introduces the SU(2) formalism, defining angular momentum states and operators, and investigates the SU(2) formalism in the harmonic oscillator context and SU(2) coherent states and their resolution of identity. Analytical representations based on SU(2) coherent states are developed and numerically implemented in the extended complex plane. The investigation explores time evolution in periodic and non-periodic systems, examining special cases like joining two paths and the behavior of zeros of the analytic representation function approaching zero and ±∞. Furthermore, the research explores Dirac contour representation in a finite dimensional Hilbert space in the extended complex plane. It defines the scalar product of states and proves the relationship between Dirac ket and Dirac bra representations. Time evolution in periodic and non-periodic systems is investigated with numerical implementations and visualizations using MATLAB. Additionally, this study applies Dirac contour representation to finite-dimensional Hilbert spaces at positive and negative temperatures, denoted as H(2j+1)+ and H(2j+1)−, respectively. It defines operators within these spaces and introduces the scalar product of two states in each system. The investigation explores L operators capable of mapping states from H(2j+1)+ to H(2j+1)− and vice versa, examining their Dirac contour representation and obtaining intriguing numerical results visualized using MATLAB.
A quantum statistical formalism which accommodates both positive and negative temperatures is studied. The Dirac contour representation is used to extend the harmonic oscillator Hubert space into a larger space which is suitable for the description of quantum systems at both positive and negative temperatures. The analytic continuation of various physical quantities into the negative temperature region is examined in detail. The formalism can be useful for the description of systems which are excited to higher states, and which decay into the lower states.
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This study considers the finite-dimensional Hilbert space with the quantum systems. TheSU(2)SU(2)SU(2)coherent states in the extended complex plane, which is stereo graphically equivalent to a sphere, serve as the foundation for analytical representation functions of the states in this space. The analytical representation function is practically implemented using the numerical style for both the periodic and non-periodic systems. This study shows in detail the zeroes trajectories of these analytic representation functions which apply the chronological evolution of the system and derive vital findings regarding to the behaviour of the quantum system. More potentially, there are several crucial applications of quantum mechanics to advance energy works for more in depth investigation in future pathways.
A contour representation in the unit disk based on SU(1,1) coherent states is introduced. The scalar product is given by a contour integral. The regions of convergence of the functions representing ket and bra states are studied. An analytic representation in the unit disk is also considered, where the scalar product is represented by an integral over the unit disk, with the Lobachevsky measure. Various relations which connect these analytic functions with other phase-space quantities are derived.
We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.