2000 character limit reached
Orthosymplectic diagonalization in Williamson's theorem (2403.11609v2)
Published 18 Mar 2024 in math.FA, math-ph, math.MP, math.SP, and quant-ph
Abstract: In this paper, we provide an algebraic condition on any $2n \times 2n$ real symmetric positive definite matrix which is necessary and sufficient for the matrix to be diagonalized by an orthosymplectic matrix in the sense of Williamson's theorem.
- John Williamson. On the algebraic problem concerning the normal forms of linear dynamical systems. American Journal of Mathematics, 58(1):141–163, 1936.
- V. I. Arnold. Mathematical Methods of Classical Mechanics. Springer New York, 1989.
- The real symplectic groups in quantum mechanics and optics. Pramana, 45:471–497, December 1995. arXiv:quant-ph/9509002.
- Maurice A. de Gosson. Symplectic Geometry and Quantum Mechanics, volume 166. Springer Science & Business Media, 2006.
- Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, 2012.
- Extremal entanglement and mixedness in continuous variable systems. Physical Review A, 70(2):022318, August 2004. arXiv:quant-ph/0402124.
- Xiao-yu Chen. Gaussian relative entropy of entanglement. Physical Review A, 71(6):062320, June 2005. arXiv:quant-ph/0402109.
- K. R. Parthasarathy. Symplectic dilations, Gaussian states and Gaussian channels. Indian Journal of Pure and Applied Mathematics, 46:419–439, August 2015. arXiv:1405.6476.
- F. Nicacio. Williamson theorem in classical, quantum, and statistical physics. American Journal of Physics, 89(12):1139–1151, December 2021. arXiv:2106.11965.
- Entanglement dynamics of coupled quantum oscillators in independent nonMarkovian baths. Entropy, 24(12):1814, December 2022. arXiv:2211.07124.
- Symplectic decomposition from submatrix determinants. Proceedings of the Royal Society A, 477(2255):20210513, November 2021. arXiv:2108.05364.
- On symplectic eigenvalues of positive definite matrices. Journal of Mathematical Physics, 56(11):112201, November 2015. arXiv:1803.04647.
- Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter. Linear Algebra and its Applications, 553:129–144, September 2018. arXiv:1710.00494.
- Hemant K. Mishra. First order sensitivity analysis of symplectic eigenvalues. Linear Algebra and its Applications, 604:324–345, November 2020. arXiv:2007.10572.
- A Schur-Horn theorem for symplectic eigenvalues. Linear Algebra and its Applications, 599:133–139, August 2020. arXiv:2004.03906.
- Variational principles for symplectic eigenvalues. Canadian Mathematical Bulletin, 64(3):553–559, September 2021.
- Tanvi Jain. Sums and products of symplectic eigenvalues. Linear Algebra and its Applications, 631:67–82, December 2021. arXiv:2108.10741.
- Derivatives of symplectic eigenvalues and a Lidskii type theorem. Canadian Journal of Mathematics, 74(2):457–485, April 2022. arXiv:2004.11024.
- Paul-Emile Paradan. The Horn cone associated with symplectic eigenvalues. Comptes Rendus. Mathématique, 360:1163–1168, May 2022. arXiv:2202.10260.
- Block perturbation of symplectic matrices in Williamson’s theorem. Canadian Mathematical Bulletin, 67(1):201–214, March 2024. arXiv:2307.01078.
- Computing symplectic eigenpairs of symmetric positive-definite matrices via trace minimization and Riemannian optimization. SIAM Journal on Matrix Analysis and Applications, 42(4):1732–1757, December 2021. arXiv:2101.02618.
- Shaowu Huang. A new version of Schur–Horn type theorem. Linear and Multilinear Algebra, 71(1):41–46, January 2023.
- Real normal operators and Williamson’s normal form. Acta Scientiarum Mathematicarum, 85:507–518, December 2019. arXiv:1804.03921.
- An order relation between eigenvalues and symplectic eigenvalues of a class of infinite dimensional operators. June 2023. arXiv:2212.03900.
- Hemant K. Mishra. Equality in some symplectic eigenvalue inequalities. arXiv preprint, 2024. arXiv:2309.04562.
- Computing symplectic eigenpairs of symmetric positive-definite matrices via trace minimization and Riemannian optimization. SIAM Journal on Matrix Analysis and Applications, 42(4):1732–1757, December 2021.
- Matrix Analysis. Cambridge University Press, 2012. Second Edition.