Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

• URL: http://arxiv.org/abs/2005.11497v1
• Date: Sat, 23 May 2020 09:11:51 GMT
• Title: Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States
• Authors: Julio A. L\'opez-Sald\'ivar, Margarita A. Man'ko, Vladimir I. Man'ko
• Abstract summary: We study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schr"odinger equation.
• Score: 1.2891210250935143
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian $\hat{H}$. We also present the corresponding differential equations which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schr\"odinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.

Related papers

• Efficient conversion from fermionic Gaussian states to matrix product states [48.225436651971805]
  We propose a highly efficient algorithm that converts fermionic Gaussian states to matrix product states. It can be formulated for finite-size systems without translation invariance, but becomes particularly appealing when applied to infinite systems. The potential of our method is demonstrated by numerical calculations in two chiral spin liquids.
  arXiv  Detail & Related papers  (2024-08-02T10:15:26Z)
• Relaxing Continuous Constraints of Equivariant Graph Neural Networks for Physical Dynamics Learning [39.25135680793105]
  We propose a general Discrete Equivariant Graph Neural Network (DEGNN) that guarantees equivariance to a given discrete point group. Specifically, we show that such discrete equivariant message passing could be constructed by transforming geometric features into permutation-invariant embeddings. We show that DEGNN is data efficient, learning with less data, and can generalize across scenarios such as unobserved orientation.
  arXiv  Detail & Related papers  (2024-06-24T03:37:51Z)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• Non-Hermitian Hamiltonian Deformations in Quantum Mechanics [4.071207179756646]
  We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we discuss the spectral properties of the Liouvillians.
  arXiv  Detail & Related papers  (2022-11-10T09:25:59Z)
• Gauge- and coordinate-invariant equations for two-component systems [0.0]
  equations are derived in a form that is invariant to gauge and coordinate transformations. equations equivalent to the nonrelativistic Schr"odinger equation of a molecule are derived as an example.
  arXiv  Detail & Related papers  (2022-07-05T17:57:43Z)
• Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras [0.0]
  We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of general size $N$. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realised in these matrix systems with permutation symmetry.
  arXiv  Detail & Related papers  (2022-07-05T16:47:10Z)
• On the Kullback-Leibler divergence between pairwise isotropic Gaussian-Markov random fields [93.35534658875731]
  We derive expressions for the Kullback-Leibler divergence between two pairwise isotropic Gaussian-Markov random fields. The proposed equation allows the development of novel similarity measures in image processing and machine learning applications.
  arXiv  Detail & Related papers  (2022-03-24T16:37:24Z)
• Learning Equivariant Energy Based Models with Equivariant Stein Variational Gradient Descent [80.73580820014242]
  We focus on the problem of efficient sampling and learning of probability densities by incorporating symmetries in probabilistic models. We first introduce Equivariant Stein Variational Gradient Descent algorithm -- an equivariant sampling method based on Stein's identity for sampling from densities with symmetries. We propose new ways of improving and scaling up training of energy based models.
  arXiv  Detail & Related papers  (2021-06-15T01:35:17Z)
• Gaussian Continuous-Variable Isotropic State [0.0]
  We study the non-classical correlations contained in a two-mode Gaussian analogue of an isotropic state. It turns out that it exhibits an analogous phenomenology as the finite-dimensional two-qubit isotropic state.
  arXiv  Detail & Related papers  (2021-05-07T09:45:15Z)
• Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of $\hbar$ [0.0]
  We show that the method developed by Gangopadhyaya, Mallow, and their coworkers can be generalized to deformed shape invariant potentials in supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to $hbar$-independent superpotentials and to a superpotential explicitly depending on $hbar$.
  arXiv  Detail & Related papers  (2020-09-30T09:55:38Z)
• Inverse Learning of Symmetries [71.62109774068064]
  We learn the symmetry transformation with a model consisting of two latent subspaces. Our approach is based on the deep information bottleneck in combination with a continuous mutual information regulariser. Our model outperforms state-of-the-art methods on artificial and molecular datasets.
  arXiv  Detail & Related papers  (2020-02-07T13:48:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.