Some Aspects of Operator Algebras in Quantum Physics

Reyes-Lega, A. F.

doi:10.1142/9789814730884_0001

Cited by 4 publications

(4 citation statements)

References 47 publications

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“…To take advantage of the symmetry between z and z −1 , we rewrite (22) in terms of Chebyshev polynomials:…”

Section: Proposition 6 (The Characteristic Polynomial Ofmentioning

confidence: 99%

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Eigenvalues of the laplacian matrices of the cycles with one weighted edge

Grudsky¹,

Maximenko²,

Alejandro³

2022

Preprint

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In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α and the others of weight 1. We denote by n the order of the graph and suppose that n tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on Re(α). After that, through the rest of the paper we suppose that 0 < α < 1. It is easy to see that the eigenvalues belong to [0, 4] and are asymptotically distributed as the function g(x) = 4 sin 2 (x/2) on [0, π]. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of [0,4]. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every n ≥ 3. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue.

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“…To take advantage of the symmetry between z and z −1 , we rewrite (22) in terms of Chebyshev polynomials:…”

Section: Proposition 6 (The Characteristic Polynomial Ofmentioning

confidence: 99%

“…Over the past decade, there has been an increasing interest in Toeplitz matrices with certain perturbations, see [3,7,8,11,12,14,21,22,27,28,32], or [17,20,23,29] for more general researches. In [11,12] the authors find the characteristic polynomial for some cases of Toeplitz matrices with corner perturbations.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

Grudsky¹,

Maximenko²,

Alejandro³

2022

Preprint

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“…Matrices L α,n can be considered as tridiagonal Toeplitz matrices with perturbations in the corners (1,1), (1, n), (n, 1) and (n, n). Several investigations in this area and some of its applications have been recently developed, see for example [2][3][4]6,7,10,12,13,15,16,[20][21][22]. These matrices can also be considered as periodic Jacobi matrices.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

Grudsky

Maximenko

Alejandro

2022

Linear Algebra and its Applications

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“…A state ω on A is then defined as a normalized positive linear functional ω : A → C. If a = a * ∈ A is an observable, ω(a) is the expectation value of a in the state ω. A readable exposition of this point of view can be found in [15,16,20].…”

mentioning

confidence: 99%

Emergent gauge symmetries and quantum operations

Balachandran

Burbano

Reyes-Lega

et al. 2020

J. Phys. A: Math. Theor.

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The algebraic approach to quantum physics emphasizes the role played by the structure of the algebra of observables and its relation to the space of states. An important feature of this point of view is that subsystems can be described by subalgebras, with partial trace being replaced by the more general notion of restriction to a subalgebra. This, in turn, has recently led to applications to the study of entanglement in systems of identical particles. In the course of those investigations on entanglement and particle identity, an emergent gauge symmetry has been found by Balachandran, de Queiroz and Vaidya. In this letter we establish a novel connection between that gauge symmetry, entropy production and quantum operations. Thus, let A be a system described by a finite dimensional observable algebra and ω a mixed faithful state. Using the Gelfand-Naimark-Segal (GNS) representation we construct a canonical purification of ω, allowing us to embed A into a larger system C. Using Tomita-Takasaki theory, we obtain a subsystem decomposition of C into subsystems A and B, without making use of any tensor product structure.We identify a group of transformations that acts as a gauge group on A while at the same time giving rise to entropy increasing quantum operations on C. We provide physical means to simulate this gauge symmetry/quantum operation duality.

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Some Aspects of Operator Algebras in Quantum Physics

Cited by 4 publications

References 47 publications

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

Emergent gauge symmetries and quantum operations

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