Related papers: Symplectic decomposition from submatrix determinan…

We consider Gaussian states of fermionic systems and study the action of the partial transposition on the density matrix. It is shown that, with a suitable choice of basis, these states are transformed into a linear combination of two…

A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can find a unitary transformation such that the transformed quadratic Hamiltonian is diagonal but still written in terms of the original position and…

This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality.…

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex…

The eigenvalues and eigenvectors of the Fisher information matrix (FIM) can reveal the most and least sensitive directions of a system and it has wide application across science and engineering. We present a symplectic variant of the…

In this paper we propose a perturbative method for the reconstruction of the covariance matrix of a multinormal distribution, under the assumption that the only available information amounts to the covariance matrix of a spherically…

Coupling qubits together towards large-scale integration is a key point for realizing a quantum computer. We study the capacitively coupled superconducting phase qubits using two diagonalization methods, which are very efficient to obtain…

Algorithms for Gaussian process, marginal likelihood methods or restricted maximum likelihood methods often require derivatives of log determinant terms. These log determinants are usually parametric with variance parameters of the…

Gaussian latent variable models are a key class of Bayesian hierarchical models with applications in many fields. Performing Bayesian inference on such models can be challenging as Markov chain Monte Carlo algorithms struggle with the…

Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. It uses projection onto a set of basis states which are typically not orthogonal. One needs to invert a matrix whose…

The paper gives a symplectic-geometric account of semiclassical Gaussian wave packet dynamics. We employ geometric techniques to "strip away" the symplectic structure behind the time-dependent Schr\"odinger equation and incorporate it into…

A basic problem in linear particle optics is to find a symplectic transformation that brings the (symmetric) beam matrix to a special diagonal form, called normal form. The conventional way to do this involves an eigenvalue-decomposition of…

The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices due to Heisenberg's uncertainty principle. This has the…

For any real diagonalizable matrix with complex eigenvalues we provide a real, coordinate free decomposition with a clear geometric interpretation.

We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an…

This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states. Such quadratic-exponential functionals…

Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present…

We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive…

In this paper we prove that there exists an asymptotical diagonalization algorithm for a class of sparse Hermitian (or real symmetric) matrices if and only if the matrices become Hessenberg matrices after some permutation of rows and…

In a differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and…