# Perturbation Bounds for Williamson's Symplectic Normal Form

@article{Idel2016PerturbationBF, title={Perturbation Bounds for Williamson's Symplectic Normal Form}, author={Martin Idel and S. Gaona and M. Wolf}, journal={arXiv: Spectral Theory}, year={2016} }

Given a real-valued positive semidefinite matrix, Williamson proved that it can be diagonalised using symplectic matrices. The corresponding diagonal values are known as the symplectic spectrum. This paper is concerned with the stability of Williamson's decomposition under perturbations. We provide norm bounds for the stability of the symplectic eigenvalues and prove that if $S$ diagonalises a given matrix $M$ to Williamson form, then $S$ is stable if the symplectic spectrum is nondegenerate… Expand

#### 8 Citations

Real Normal Operators and Williamson's Normal Form

- Mathematics
- 2018

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric… Expand

Symplectic eigenvalue problem via trace minimization and Riemannian optimization

- Mathematics
- 2021

We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson’s theorem. It is… Expand

First order sensitivity analysis of symplectic eigenvalues

- Mathematics
- 2020

Abstract For every 2 n × 2 n positive definite matrix A there are n positive numbers d 1 ( A ) ≤ … ≤ d n ( A ) associated with A called the symplectic eigenvalues of A. It is known that d m are… Expand

Optimal gauge for the multimode Rabi model in circuit QED

- Physics
- 2019

In circuit QED, a Rabi model can be derived by truncating the Hilbert space of the anharmonic qubit coupled to a linear, reactive environment. This truncation breaks the gauge invariance present in… Expand

Continuous-variable ramp quantum secret sharing with Gaussian states and operations

- Physics, Computer Science
- New Journal of Physics
- 2019

A technique for certifying continuous-variable ramp quantum secret-sharing schemes in the framework of quantum interactive-proof systems and derives the expression for quantum mutual information between the quantum secret extracted by any multi-player structure and the share held by the referee corresponding to the Tyc-Rowe-Sanders continuous- variables quantum secret sharing scheme. Expand

Optique quantique multimode pour le traitement de l'information quantique

- Physics
- 2019

Cette these etudie l’optique quantique multimode, aussi bien du point de vue de la generation que celui de la detection. Elle s’articule autour de trois volets. Nous etudions la generation de lumiere… Expand

Derivatives of symplectic eigenvalues and a Lidskii type theorem

- Mathematics
- 2020

Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist $n$ positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb{R}^{2n}$ called the… Expand

Free-mode removal and mode decoupling for simulating general superconducting quantum circuits

- Computer Science, Physics
- 2020

This work considers and solves two issues involved in simulating general superconducting circuits, including the handling of free modes in the circuit and the challenge of simulating large circuits, by giving a provably correct algorithm for removing free modes by performing a linear canonical transformation. Expand

#### References

SHOWING 1-10 OF 20 REFERENCES

On symplectic eigenvalues of positive definite matrices

- Mathematics, Physics
- 2015

If A is a 2n × 2n real positive definite matrix, then there exists a symplectic matrix M such that MTAM=DOOD where D = diag(d1(A), …, dn(A)) is a diagonal matrix with positive diagonal entries, which… Expand

Congruences and canonical forms for a positive matrix: Application to the Schweinler–Wigner extremum principle

- Mathematics, Physics
- 1999

It is shown that a N×N real symmetric [complex Hermitian] positive definite matrix V is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in SO(m,n)[SU(m,n)], for any… Expand

On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems

- Mathematics
- 1936

Introduction. Let m be the number of degrees of freedom of a linear conservative dynamical systenm and let the point (q1, q2,9 * , q'Mn Pl p2, . . . p'mt) of the phase space be denoted by x = (xl,… Expand

Symplectic Geometry and Quantum Mechanics

- Mathematics
- 2006

Symplectic Geometry.- Symplectic Spaces and Lagrangian Planes.- The Symplectic Group.- Multi-Oriented Symplectic Geometry.- Intersection Indices in Lag(n) and Sp(n).- Heisenberg Group, Weyl Calculus,… Expand

Gaussian entanglement of formation

- Physics
- 2004

We introduce a Gaussian version of the entanglement of formation adapted to bipartite Gaussian states by considering decompositions into pure Gaussian states only. We show that this quantity is an… Expand

Updating the Inverse of a Matrix

- Mathematics, Computer Science
- SIAM Rev.
- 1989

The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed. Expand

The conditional entropy power inequality for Gaussian quantum states

- Physics, Mathematics
- 2013

We propose a generalization of the quantum entropy power inequality involving conditional entropies. For the special case of Gaussian states, we give a proof based on perturbation theory for… Expand

Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs

- Physics
- 2005

We address the question of the multiplicativity of the maximal p-norm output purities of bosonic Gaussian channels under Gaussian inputs. We focus on general Gaussian channels resulting from the… Expand

The algebraic eigenvalue problem

- Mathematics
- 1965

Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of… Expand

Continuous Variable Quantum Information: Gaussian States and Beyond

- Mathematics, Computer Science
- Open Syst. Inf. Dyn.
- 2014

The basic notions needed to understand Gaussian states and Gaussian operations are defined, and emphasis is placed on the mathematical structure combining notions of algebra and symplectic geometry fundamental to a complete understanding of Gaussian informatics. Expand