Spectral properties of certain tridiagonal matrices

Broadcasting information anonymously becomes more difficult as surveillance technology improves, but remarkably, quantum protocols exist that enable provably traceless broadcasting. The difficulty is making scalable entangled resource states that are robust to errors. We propose an anonymous broadcasting protocol that uses a continuous-variable surface-code state that can be produced using current technology. High squeezing enables large transmission bandwidth and strong anonymity, and the topological nature of the state enables local error mitigation.

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“…This result also agrees with the literature [68][69][70][71] after applying properties of Chebyshev polynomials. Define c n (x) det C n (x).…”

Section: Preparation By Measurement Of a CV Cluster Statesupporting

confidence: 92%

Anonymous broadcasting of classical information with a continuous-variable topological quantum code

et al. 2018

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“…where U n (cos k) are the Chebyshev polynomials of the second kind defined in (24). From the characteristic equation for the whole system, W 1 N (λ, cos k) = 0, one arrives at (26)(27) with…”

Section: General Methodsmentioning

confidence: 99%

“…Since φ 1,e λ,± (p) in (37) has to follow directly from φ 1 λ (k) in (27) (31), as will be shown in Section V A. By further applying η λ (k) → η e λ,± (p) to (31), with η = a, b, c, d, the eigenstates of the edge states are written as, apart from a global phase factor,…”

Section: A Edge Statesmentioning

confidence: 99%

“…Central to the studies carried out below is that the Hamiltonian of an open 1D linear model with arbitrary on-site potentials v i and hopping constants t i within the unit cell, labeled ionic SSH m (ISSH m ), where m is the size of the unit cell (see figure 1), is a periodic Hermitian block-tridiagonal matrix [24], also called a tridiagonal Hermitian m-Toeplitz matrix [25][26][27][28][29]. Each ISSH m model corresponds to an AAH model with specific periodic modulations on t i and v i , which can be different in general, labeled here as commensurate AAH model.…”

Section: Introductionmentioning

confidence: 99%

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Analytical solution of open crystalline linear 1D tight-binding models

Marques

Dias

2020

J. Phys. A: Math. Theor.

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A method for finding the exact analytical solutions for the bulk and edge energy levels and corresponding eigenstates for all commensurate Aubry-André/Harper single-particle models under open boundary conditions is presented here, both for integer and non-integer number of unit cells. The solutions are ultimately found to be dependent on the behavior of phase factors whose compact formulas, provided here, make this method simple to implement computationally. The derivation employs the properties of the Hamiltonians of these models, all of which can be written as Hermitian block-tridiagonal Toeplitz matrices. The concept of energy spectrum is generalized to incorporate both bulk and edge bands, where the latter are a function of a complex momentum. The method is then extended to solve the case where one of these chains is coupled at one end to an arbitrary cluster/impurity. Future developments based on these results are discussed.

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“…However simple these systems appear, however important they are and for however long they have been studied, they are still under investigation. The present contribution is notably closely related to recent work on analytical expressions of the eigenvalues, eigenvectors and inverse of tridiagonal matrices that have two or four of their corner coefficients disturbed [16][17][18][19][20][21]. The focus here is on the location of not only poles but also zeros of the transfer functions of a disturbed system in the frequency-disturbance magnitude plane.…”

Section: Introductionmentioning

confidence: 91%

Zero and root loci of disturbed spring–mass systems

Lecomte

2014

Proc. R. Soc. A.

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Models consisting of chains of particles that are coupled to their neighbours appear in many applications in physics or engineering, such as in the study of dynamics of mono-atomic and multi-atomic lattices, the resonances of crystals with impurities and the response of damaged bladed discs. Analytical properties of the dynamic responses of such disturbed chains of identical springs and masses are presented, including when damping is present. Several remarkable properties in the location of the resonances (poles) and anti-resonances (zeros) of the displacements in the frequency domain are presented and proved. In particular, it is shown that there exists an elliptical region in the frequency–disturbance magnitude plane from which zeros are excluded and the discrete values of the frequency and disturbance at which double poles occur are identified. A particular focus is on a local disturbance, such as when a spring or damper is modified at or between the first and last masses. It is demonstrated how, notably through normalization, the techniques and results of the paper apply to a broad category of more complex systems in physics, chemistry and engineering.

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Spectral properties of certain tridiagonal matrices

Abstract: We study spectral properties of irreducible tridiagonal k−Toeplitz matrices and certain matrices which arise as perturbations of them.

Cited by 7 publications

References 23 publications

Anonymous broadcasting of classical information with a continuous-variable topological quantum code

Anonymous broadcasting of classical information with a continuous-variable topological quantum code

Analytical solution of open crystalline linear 1D tight-binding models

Zero and root loci of disturbed spring–mass systems

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