A structure-preserving algorithm for the quaternion Cholesky decomposition

Wang, Minghui; Ma, Wenhao

doi:10.1016/j.amc.2013.08.026

Cited by 14 publications

(3 citation statements)

References 22 publications

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“…(Note four "active" variables in the first integrand (37), and only two radial and no angular ones in the second (40).) The integration constraints (38) now simply reduced to r 2 13 + r 2 14 µ 2 < 1, with µ > 1.…”

Section: B Eleven-dimensional Convex Set Of Two-qubit Statesmentioning

confidence: 99%

“…We wanted to test the fit of these last two functions by the generation of random two-quaterbit matrices-but had not yet found an effective manner of doing so (cf. [38][39][40]). (The possible use of Ginibre ensembles, in the manner of [38], and the associated issues in doing so, has been addressed by C. F. Dunkl in App.…”

Section: Attempted Construction Of χ4 (ε)mentioning

confidence: 99%

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Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures

Slater

2017

Preprint

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We begin by investigating relationships between two forms of Hilbert-Schmidt two-rebit and two-qubit "separability functions"-those recently advanced by Lovas and Andai (J. Phys. A 50[2017] 295303), and those earlier presented by Slater (J. Phys. A 40 [2007] 14279). In the Lovas-Andai framework, the independent variable ε ∈ [0, 1] is the ratio σ(V ) of the singular values offormed from the two 2 × 2 diagonal blocks (D 1 , D 2 ) of a 4 × 4 density matrix D = ρ ij . In the Slater setting, the independent variable µ is the diagonal-entry ratio ρ 11 ρ 44 ρ 22 ρ 33 -with, of central importance, µ = ε or µ = 1 ε when both D 1 and D 2 are themselves diagonal. Lovas and Andai established that their two-rebit "separability function" χ1 (ε) (≈ ε) yields the previously conjectured Hilbert-Schmidt separability probability of 29 64 . We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we newly find its two-qubit, two-quater[nionic]-bit and "two-octo[nionic]-bit" counterparts, χ2 (εThese immediately lead to predictions of Hilbert-Schmidt separability/PPT-probabilities of 8 33 , 26 323 and 44482 4091349 , in full agreement with those of the "concise formula" (J. Phys. A 46 [2013] 445302), and, additionally, of a "specialized induced measure" formula. Then, we find a Lovas-Andai "master formula", χd (ε) =+1) 2 , encompassing both even and odd values of d. Remarkably, we are able to obtain the χd (ε) formulas, d = 1, 2, 4, applicable to full (9-, 15-, 27-) dimensional sets of density matrices, by analyzing (6-, 9, 15-) dimensional sets, with not only diagonal D 1 and D 2 , but also an additional pair of nullified entries. Nullification of a further pair still, leads to X-matrices, for which a distinctly different, simple Dyson-index phenomenon is noted. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence. A two-qubit separability probability of 1 − 256 27π 2 is obtained based on the operator monotone function √ x, with the use of χ2 (ε).

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Section: B Eleven-dimensional Convex Set Of Two-qubit Statesmentioning

confidence: 99%

Section: Attempted Construction Of χ4 (ε)mentioning

confidence: 99%

Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures

Slater

2017

Preprint

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“…In the next, we will reformulate the problem in (28), so that it can be solved by SOC programming based method.…”

Section: Quaternion-valued Worst-case Constrained Algorithmmentioning

confidence: 99%

Quaternion-valued robust adaptive beamformer for electromagnetic vector-sensor arrays with worst-case constraint

Zhang

Liu

et al. 2014

Signal Processing

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A robust adaptive beamforming scheme based on two-component electromagnetic (EM) vector-sensor arrays is proposed by extending the well-known worst-case constraint into the quaternion domain. After defining the uncertainty set of the desired signal's quaternionic steering vector, two quaternionvalued constrained minimization problems are derived. We then reformulate them into two real-valued convex quadratic problems, which can be easily solved via the so-called second-order cone (SOC) programming method.It is also demonstrated that the proposed algorithms can be classified as a specific type of the diagonal loading scheme, in which the optimal loading factor is a function of the known level of uncertainty of the desired steering vector. Numerical simulations show that our new method can cope with the steering vector mismatch problem well, and alleviate the finite sample * Corresponding author: Tel.: +44-114-2225813; fax: +44-114-2225834.Email addresses: xrzhang@bit.edu.cn (Xirui Zhang), w.liu@sheffield.ac.uk (Wei Liu), yougenxu@bit.edu.cn (Yougen Xu), zwliu@bit.edu.cn (Zhiwen Liu) February 27, 2014 size effect to some extent. Besides, the proposed beamformer significantly outperforms the sample matrix inversion minimum variance distortionless response (SMI-MVDR) and the quaternion Capon (Q-Capon) beamformers in all the scenarios studied, and achieves a better performance than the traditional diagonal loading scheme, in the case of smaller sample sizes and higher SNRs. Preprint submitted to Signal Processing

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A complex structure-preserving algorithm for split quaternion matrix LDU decomposition in split quaternion mechanics

Wang

Jiang

Guo

et al. 2021

Calcolo

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A structure-preserving algorithm for the quaternion Cholesky decomposition

Cited by 14 publications

References 22 publications

Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures

Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures

Quaternion-valued robust adaptive beamformer for electromagnetic vector-sensor arrays with worst-case constraint

A complex structure-preserving algorithm for split quaternion matrix LDU decomposition in split quaternion mechanics

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