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Symmetry

Algebraic and Spectral Analysis of a Novel Hermitian Spin Basis

Timothy Ganesan, Z. Yousaf, Muhammad Zaeem Ul Haq Bhatti

Published Mar 17, 2025• 0 citations
Publisher

Abstract

This work aims to explore the algebraic and spectral properties of a novel parametric Hermitian non-Pauli spin basis. The primary motivation for this work is to introduce an alternative to the Pauli spin basis for investigating systems where conventional quantum statistics no longer apply. This study explores the symmetries, commutation relations, Lie algebra structures, and ladder operators of the proposed spin matrices. Additionally, it provides discussions on key findings, potential applications, and concludes with some final remarks. An example for modeling the spectrum of a quantum system is provided.

DOI
10.3390/sym17030450

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This work aims to explore the algebraic and spectral properties of a novel parametric Hermitian non-Pauli spin basis. The primary motivation for this work is to introduce an alternative to the Pauli spin basis for investigating systems where conventional quantum statistics no longer apply. This study explores the symmetries, commutation relations, Lie algebra structures, and ladder operators of the proposed spin matrices. Additionally, it provides discussions on key findings, potential applications, and concludes with some final remarks. An example for modeling the spectrum of a quantum system is provided.
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Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class A\mathbf{A}A for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.
In this work we present a pedagogically motivated analysis of spin-correlation calculations in a quantum system composed of two spin-1/21/21/2 particles. Rather than aiming at new physical results, our purpose is to clarify and bring attention to different strategies for evaluating expectation values of the form ⟨ψ∣Su^(1)Sv^(2)∣ψ⟩\langle ψ| S^{(1)}_{\hat{\boldsymbol{u}}} S^{(2)}_{\hat{\boldsymbol{v}}} | ψ\rangle⟨ψ∣Su^(1)​Sv^(2)​∣ψ⟩, which play an important role in discussions of entanglement and Bell-type correlations. We compare three complementary approaches. The first follows a direct algebraic evaluation in the product basis, closely related to standard textbook methods. The second uses a matrix representation of bipartite states, in which the tensor-product structure is expressed in terms of 2×22\times22×2 complex matrices. This representation keeps the calculation close to the familiar Pauli-matrix algebra and makes the independent action of operators on each subsystem more transparent. The third explores a symmetry-based argument, highlighting both its usefulness and its limitations when applied beyond the singlet state. We show explicitly that the singlet state is rotationally invariant, which explains why the symmetry argument successfully reproduces its correlation function, while a naive extension fails for triplet states. The discussion illustrates how entanglement, tensor-product structure, and rotational symmetry interplay in spin correlations.
The evolution of quantum states serves as good fundamental studies in understanding the quantum information systems which finally lead to the research on quantum computation. To carry out such a study, mathematical tools such as the Lie group and their associated Lie algebra is of great importance. In this study, the Lie algebra of su(8) is represented in a tensor product operation between three Pauli matrices. This can be realized by constructing the generalized Gell-Mann matrices and comparing them to the Pauli bases. It is shown that there is a one-to-one correlation of the Gell-Mann matrices with the Pauli basis which resembled the change of coordinates. Together with the commutator relations and the frequency analysis of the structure constant via the algebra, the Lie bracket operation will be highlighted providing insight into relating quantum circuit model with Lie Algebra. These are particularly useful when dealing with three-qubit quantum circuit problems which involve quantum gates that is derived from the SU(8) Lie group.