8 research outputs found
Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic
It was shown in a series of recent publications that the eigenvalues of
Toeplitz matrices generated by so-called simple-loop symbols admit
certain regular asymptotic expansions into negative powers of . On the
other hand, recently two of the authors considered the pentadiagonal Toeplitz
matrices generated by the symbol , which does not satisfy
the simple-loop conditions, and derived asymptotic expansions of a more
complicated form. We here use these results to show that the eigenvalues of the
pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic
expansion. This also delivers a counter-example to a conjecture by Ekstr\"{o}m,
Garoni, and Serra-Capizzano and reveals that the simple-loop condition is
essential for the existence of the regular asymptotic expansion.Comment: 28 pages, 7 figure
Relationships between the permanents of a certain type of k-tridiagonal symmetric Toeplitz matrix and the Chebyshev polynomials
In this study, the recursive relations between the permanents of a certain type of the k-tridiagonal symmetric Toeplitz matrix with complex entries and the Chebyshev polynomials of the second kind are presented
Exact equivalences and phase discrepancies between random matrix ensembles
We study two types of random matrix ensembles that emerge when considering
the same probability measure on partitions. One is the Meixner ensemble with a
hard wall and the other are two families of unitary matrix models, with weight
functions that can be interpreted as characteristic polynomial insertions. We
show that the models, while having the same exact evaluation for fixed values
of the parameter, may present a different phase structure. We find phase
transitions of the second and third order, depending on the model. Other
relationships, via direct mapping, between the unitary matrix models and
continuous random matrix ensembles on the real line, of Cauchy-Romanovski type,
are presented and studied both exactly and asymptotically. The case of
orthogonal and symplectic groups is studied as well and related to Wronskians
of Chebyshev polynomials, that we evaluate at large .Comment: 41 pages, 10 figures. v2: some explanations and references added,
final versio
Exact equivalences and phase discrepancies between random matrix ensembles
We study two types of random matrix ensembles that emerge when considering the same probability measure on partitions. One is the Meixner ensemble with a hard wall and the other are two families of unitary matrix models, with weight functions that can be interpreted as characteristic polynomial insertions. We show that the models, while having the same exact evaluation for fixed values of the parameter, may present a different phase structure. We find phase transitions of the second and third order, depending on the model. Other relationships, via direct mapping, between the unitary matrix models and continuous random matrix ensembles on the real line, of Cauchy-Romanovski type, are presented and studied both exactly and asymptotically. The case of orthogonal and symplectic groups is studied as well and related to Wronskians of Chebyshev polynomials, that we evaluate at largeN.info:eu-repo/semantics/acceptedVersio
Anonymous broadcasting of classical information with a continuous-variable topological quantum code
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