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Entropy

Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States

Julio A. López-Saldívar, Olga V. Man’ko, Margarita A. Man’ko, Vladimir I. Man’ko

Published Dec 6, 2023• 2 citations

Abstract

The Jordan–Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan–Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

DOI: 10.3390/e25121628

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The Jordan–Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan–Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

Introduction The space of turns was shown in the preceding chapter to be the carrier space of the defining (fundamental) representation of the group SU (2) * SU (2) – the group of the symmetric top. We shall show in the present chapter how the complete representation theory of this structure can be obtained from a mapping (the generalized Jordan map) that maps the generic turn into an operator (the matrix boson operator) acting in a Hilbert space. By means of this construction we shall obtain a unified presentation of the angular momentum states, the rotation matrices, and the Wigner coefficients as well. An interesting aspect of this technique is that it leads automatically to a view of the Wigner coefficients as a form of discretized rotation matrix (Section 8 below), which in turn implies a relationship between the Wigner coefficients and Jacobi polynomials (Gel'fand et al . [1, Supplement III]). The key idea behind this unified presentation – the Jordan mapping of the turn into a matrix boson – is of more general validity than the application made here. To put this structure in a larger context we first develop, in some detail, the techniques of the boson calculus . It is hoped that this digression does not break the thread of the development too badly. Excursus on the Boson Calculus The harmonic oscillator . The boson calculus originated in the treatment of a basic physical problem, the quantum mechanics of the linear harmonic oscillator.

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

We show the use of the theory of Lie algebras, especially their oscillator realizations, in the context of quantum mechanics. One can construct oscillator realizations from matrix realizations. In the case of symplectic and special orthogonal algebra, we demonstrate an alternative method of obtaining oscillator realizations from symmetric or exterior power of a vector space of annihilation and creation bosonic or fermionic operators. We find Lie algebra of polynomials of degree at most two in phase space of a mechanical system, which form the semi-direct product of the Heisenberg algebra and symplectic algebra. It is shown that a classical system with Hamiltonian function in this algebra can be quantized by two equivalent representations - Schrödinger or Bargmann-Fock representation. The second mentioned representation generates the same operators of symplectic algebra as we got from their previous formal construction from symmetric power of a vector space of bosonic operators. Quantization is demonstrated on the bosonic harmonic oscillator. We use the similarities between bosonic and fermionic oscillator realizations to define the fermionic harmonic oscillator. Some properties of spinor representations of special orthogonal algebra are illustrated on its state space. Powered by TCPDF (www.tcpdf.org)

The general problem of describing quantum states not only by wave functions and density operators but also by probability distribution functions is discussed in this paper. The qubit-state density-matrix elements expressed in terms of probability distributions are connected with the irreducible representation of group SU(2). The transform of the probability distributions corresponding to the unitary transform of the qubit state of the spin-1/2 system is obtained and expressed in terms of matrix elements of the group SU(2). The possibility to extend the introduced formalism to other quantum states is suggested. For qubit systems, the notion of density operators algebra and the relation with the probability representation of quantum states are discussed. The Schrödinger equation for qubit states is obtained as an equation for probabilities.

It is shown that a SU(1,1) algebra may be used to provide a unified description of the simple hamonic oscillator and the angular momentum algebras and a class of other semi-infinite algebras. A normal ordered representation of a Unitary operator

U

constructed from the generators of a SU(1,1) algebra, which is a generalisation of the Baker- Campbell - Hausdorff relation for Lie algebras, is given. It is shown that the normal ordered representatiion of

U

may be used to calculate expectation values which are functions of the parameters used to construct the operator. The functions so constructed satisfy certain recurrence relations and the entire set of functions may be interpreted in terms of diagrams similar to the Pascal triangle for binomial coefficients. Coherent states, squeezed states and rotation matrices of the angular momentum algebra emerge as special cases.