scholarly journals Bosonic and fermionic Gaussian states from Kähler structures

2021 ◽  
Vol 4 (3) ◽  
Author(s):  
Lucas Hackl ◽  
Eugenio Bianchi
Keyword(s):  
Complex Structure ◽  
Covariance Matrices ◽  
Unified Framework ◽  
Linear Complex ◽  
Gaussian States ◽  
Mathematical Structures ◽  
Driven Systems ◽  
C Dynamics ◽  
Classical Phase Space ◽  
Kähler Structures

We show that bosonic and fermionic Gaussian states (also known as ``squeezed coherent states’’) can be uniquely characterized by their linear complex structure JJ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple (G,\Omega,J)(G,Ω,J) of compatible Kähler structures, consisting of a positive definite metric GG, a symplectic form \OmegaΩ and a linear complex structure JJ with J^2=-\mathbb{1}J2=−1. Mixed Gaussian states can also be identified with such a triple, but with J^2\neq -\mathbb{1}J2≠−1. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.

scholarly journals A QUANTUM IS A COMPLEX STRUCTURE ON CLASSICAL PHASE SPACE

2005 ◽  
Vol 02 (04) ◽  
pp. 633-655
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Complex Structure ◽  
Classical Mechanics ◽  
Elementary Excitation ◽  
Differentiable Structure ◽  
Duality Transformations ◽  
Kähler Potential ◽  
Classical Phase Space

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space. For that purpose we consider Kähler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kähler potential.

scholarly journals A Review of Innovation-Based Methods to Jointly Estimate Model and Observation Error Covariance Matrices in Ensemble Data Assimilation

2020 ◽  
Vol 148 (10) ◽  
pp. 3973-3994 ◽  
Author(s):  
Pierre Tandeo ◽  
Pierre Ailliot ◽  
Marc Bocquet ◽  
Alberto Carrassi ◽  
Takemasa Miyoshi ◽  
...  
Keyword(s):  
Data Assimilation ◽  
Current Data ◽  
Covariance Matrices ◽  
Estimation Methods ◽  
Observation Error ◽  
Correlated Noise ◽  
Unified Framework ◽  
Error Covariance ◽  
Error Terms ◽  
Observation Space

AbstractData assimilation combines forecasts from a numerical model with observations. Most of the current data assimilation algorithms consider the model and observation error terms as additive Gaussian noise, specified by their covariance matrices and , respectively. These error covariances, and specifically their respective amplitudes, determine the weights given to the background (i.e., the model forecasts) and to the observations in the solution of data assimilation algorithms (i.e., the analysis). Consequently, and matrices significantly impact the accuracy of the analysis. This review aims to present and to discuss, with a unified framework, different methods to jointly estimate the and matrices using ensemble-based data assimilation techniques. Most of the methods developed to date use the innovations, defined as differences between the observations and the projection of the forecasts onto the observation space. These methods are based on two main statistical criteria: 1) the method of moments, in which the theoretical and empirical moments of the innovations are assumed to be equal, and 2) methods that use the likelihood of the observations, themselves contained in the innovations. The reviewed methods assume that innovations are Gaussian random variables, although extension to other distributions is possible for likelihood-based methods. The methods also show some differences in terms of levels of complexity and applicability to high-dimensional systems. The conclusion of the review discusses the key challenges to further develop estimation methods for and . These challenges include taking into account time-varying error covariances, using limited observational coverage, estimating additional deterministic error terms, or accounting for correlated noise.

scholarly journals ON THE COHOMOLOGY OF ALMOST-COMPLEX MANIFOLDS

2012 ◽  
Vol 23 (02) ◽  
pp. 1250019 ◽  
Author(s):  
DANIELE ANGELLA ◽  
ADRIANO TOMASSINI
Keyword(s):  
Complex Structure ◽  
Almost Complex Structure ◽  
Complex Manifolds ◽  
Almost Complex Manifolds ◽  
Dynamical Study ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Kähler Structures ◽  
Foliated Manifolds ◽  
Invent Math

Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225–255] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

scholarly journals Steering Witness and Steering Criterion of Gaussian States

Entropy ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 62
Author(s):  
Ruifen Ma ◽  
Taotao Yan ◽  
Dantong Wu ◽  
Xiaofei Qi
Keyword(s):  
Continuous Variable ◽  
Covariance Matrices ◽  
Gaussian States ◽  
Quantum Steering ◽  
Bell Nonlocality ◽  
Definition Of

Quantum steering is an important quantum resource, which is intermediate between entanglement and Bell nonlocality. In this paper, we study steering witnesses for Gaussian states in continuous-variable systems. We give a definition of steering witnesses by covariance matrices of Gaussian states, and then obtain a steering criterion by steering witnesses to detect steerability of any (m+n)-mode Gaussian states. In addition, the conditions for two steering witnesses to be comparable and the optimality of steering witnesses are also discussed.

scholarly journals Unified framework to determine Gaussian states in continuous-variable systems

2017 ◽  
Vol 96 (4) ◽  
Author(s):  
Fernando Nicacio ◽  
Andrea Valdés-Hernández ◽  
Ana P. Majtey ◽  
Fabricio Toscano
Keyword(s):  
Continuous Variable ◽  
Unified Framework ◽  
Gaussian States

scholarly journals Complexity of mixed Gaussian states from Fisher information geometry

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Giuseppe Di Giulio ◽  
Erik Tonni
Keyword(s):  
Fisher Information ◽  
Circuit Complexity ◽  
Information Geometry ◽  
Covariance Matrices ◽  
Mixed States ◽  
Gaussian States ◽  
Infinite Line ◽  
Reduced Density ◽  
Special Case ◽  
Thermal States

Abstract We study the circuit complexity for mixed bosonic Gaussian states in harmonic lattices in any number of dimensions. By employing the Fisher information geometry for the covariance matrices, we consider the optimal circuit connecting two states with vanishing first moments, whose length is identified with the complexity to create a target state from a reference state through the optimal circuit. Explicit proposals to quantify the spectrum complexity and the basis complexity are discussed. The purification of the mixed states is also analysed. In the special case of harmonic chains on the circle or on the infinite line, we report numerical results for thermal states and reduced density matrices.

scholarly journals NONCOMMUTATIVE GEOMETRY ON QUANTUM PHASE SPACE

1996 ◽  
Vol 11 (07) ◽  
pp. 1253-1278 ◽  
Author(s):  
M. REUTER
Keyword(s):  
Phase Space ◽  
Differential Forms ◽  
Linear Operators ◽  
Quantum Mechanical ◽  
Classical Dynamics ◽  
Unified Framework ◽  
Tensor Fields ◽  
Classical Phase Space ◽  
Mechanical Dynamics ◽  
Arbitrary Quantum

A noncommutative analog of the classical differential forms is constructed on the phase space of an arbitrary quantum system. The noncommutative forms are universal and are related to the quantum-mechanical dynamics in the same way as the classical forms are related to classical dynamics. They are constructed by applying the Weyl-Wigner symbol map to the differential envelope of the linear operators on the quantum-mechanical Hilbert space. This leads to a representation of the noncommutative forms considered by A. Connes in terms of multiscalar functions on the classical phase space. In an appropriate coincidence limit they define a quantum deformation of the classical tensor fields and both commutative and noncommutative forms can be studied in a unified framework. We interpret the quantum differential forms in physical terms and comment on their possible applications.

scholarly journals GENERALIZED COMPLEX GEOMETRY AND THE PLANCK CONE

2006 ◽  
Vol 21 (06) ◽  
pp. 1189-1197 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Finite Number ◽  
Degrees Of Freedom ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Complex Geometry ◽  
Complex Structure ◽  
Generalized Complex ◽  
Classical Phase Space ◽  
Classical And Quantum Mechanics

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalized complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a complex structure on classical phase space. In this paper we analyze the role played by generalized complex geometry in the classical and quantum mechanics of a finite number of degrees of freedom. We identify generalized complex geometry as an appropriate geometrical setup for dualities. The latter are interpreted as transformations connecting points in the interior of the Planck cone with points in the exterior, and vice versa. The Planck cone bears some resemblance with the relativistic light-cone. However the latter cannot be traversed by physical particles, while dualities do connect the region outside the Planck cone with the region inside, and vice versa.

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