Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

Böttcher, Albrecht; Grudsky, Sergei M.; Maksimenko, E. A.

doi:10.1016/j.laa.2008.02.031

Cited by 8 publications

(3 citation statements)

References 16 publications

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“…where 𝑠 𝑗 𝑇 (𝑑) is now the 𝑗 th singular value of the matrix 𝑇 (𝑑) . Both Szegő's and Avram-Parter's result have been generalized in successive steps, by Zamarashkin and Tyrtyshnikov [107], Tilli [108], Serra-Capizzano [66], Böttcher, Grudsky, and Maksimenko [109], and others. For a detailed account of these developments, we refer the reader to the textbooks [67,110,111] and especially to the lecture notes by Grudsky [112].…”

Section: A Infinite Toeplitz Matrices and Theorems Of Szegő And Avram...mentioning

confidence: 99%

Exact solution for the quantum and private capacities of bosonic dephasing channels

Lami¹,

Wilde²

2022

Preprint

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The capacities of noisy quantum channels capture the ultimate rates of information transmission across quantum communication lines, and the quantum capacity plays a key role in determining the overhead of faulttolerant quantum computation platforms. In the case of bosonic systems, central to many applications, no closed formulas for these capacities were known for bosonic dephasing channels, a key class of non-Gaussian channels modelling, e.g., noise affecting superconducting circuits or fiber-optic communication channels. Here we provide the first exact calculation of the quantum, private, two-way assisted quantum, and secret-key agreement capacities of all bosonic dephasing channels. We prove that that they are equal to the relative entropy of the distribution underlying the channel to the uniform distribution. Our result solves a problem that has been open for over a decade, having been posed originally by [Jiang & Chen, Quantum and Nonlinear Optics 244, 2010].

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Section: A Infinite Toeplitz Matrices and Theorems Of Szegő And Avram...mentioning

confidence: 99%

Exact solution for the quantum and private capacities of bosonic dephasing channels

Lami¹,

Wilde²

2022

Preprint

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“…This problem is important for statistical mechanics and other applications ( [1], [5], [6], [7], [8]). First of all, we mention the numerous versions of the Szegö theorem on the asymptotic distribution of eigenvalues and theorems of Abram-Parter type on the asymptotic distribution of singular numbers ( [9], [10], [11], [12]). There is a rich literature devoted to the asymptotics of the determinants of Toeplitz matrices.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols

Batalshchikov¹,

Grudsky

Malisheva

et al. 2019

Linear Algebra and its Applications

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This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimension of the matrices increases to infinity. The main result describes the asymptotic structure of all eigenvalues. The constructed expansion is uniform with respect to the number of eigenvalues.

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“…Added in proof. Problems 1.2 and 2.4 were recently solved in [3]. The answer to Problem 1.2 is in the affirmative, while that to Problem 2.4 is negative.…”

Section: Higher Dimensions and Block Casementioning

confidence: 99%

Some Problems Concerning the Test Functions in the Szegö and Avram-Parter Theorems

Böttcher

Grudsky

Schwartz

Recent Advances in Operator Theory and Applications

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The Szegö and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues and singular values of Toeplitz matrices as the matrix dimension increases to infinity. This paper is concerned with some questions that arise when the test functions do not satisfy the known growth restrictions at infinity or when the test function has a logarithmic singularity within the range of the symbol. Several open problems are listed and accompanied by a few new results that illustrate the delicacy of the matter.

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Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

Cited by 8 publications

References 16 publications

Exact solution for the quantum and private capacities of bosonic dephasing channels

Exact solution for the quantum and private capacities of bosonic dephasing channels

Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols

Some Problems Concerning the Test Functions in the Szegö and Avram-Parter Theorems

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