Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Dissipative evolution of covariance matrix beyond quadratic order

Tomasz Linowski, Łukasz Rudnicki

Published May 26, 2021• 1 citation
OpenAlex

Abstract

The covariance matrix contains the complete information about the second-order expectation values of the mode quadratures (position and momentum operators) of the system. Due to its prominence in studies of continuous variable systems, most significantly Gaussian states, special emphasis is put on time evolution models that result in self-contained equations for the covariance matrix. So far, despite not being explicitly implied by this requirement, virtually all such models assume a so-called quadratic, or second-order case, in which the generator of the evolution is at most second-order in the mode quadratures. Here, we provide an explicit model of covariance matrix evolution of infinite order. Furthermore, we derive the solution, including stationary states, for a large subclass of proposed evolutions. Our findings challenge the contemporary understanding of covariance matrix dynamics and may give rise to new methods and improvements in quantum technologies employing continuous variable systems.

Related papers

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

Julio A. López-Saldívar, Margarita A. Man’ko, Vladimir I. Man’ko

In the 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 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 Schrödinger 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.

arXiv:2005.11497•Entropy•May 23, 2020•14 citations

Higher-order covariance matrices for non-Gaussian quantum states

Vojtěch Kala, Petr Marek, Nicolas J. Cerf

Covariance matrices lie at the heart of the powerful and well-established symplectic framework for describing continuous-variable Gaussian quantum states. However, since this framework only relies on first- and second-order moments, it is not sufficient for the analysis of non-Gaussian states because their higher-order moments are essential to capture some of their key properties. Here, we define higher-order covariance matrices -- more precisely, covariance matrices built from higher-order quadrature monomials -- which provide a simple way to evaluate the effect of Gaussian transformations on non-Gaussian states and can be used, for example, to address nonlinear squeezing or non-Gaussian nullifiers. Higher-order covariance matrices can be estimated from homodyne measurement data using only a limited number of quadrature angles, which involves matrices of moderate dimension compared with a full simulation in the Fock basis. The dimension of a higher-order covariance matrix does not depend on the span of the quantum states in Fock basis and, furthermore, scales only polynomially with the number of modes.

quant-pharXiv:2607.11759•Jul 14, 2026•0 citations

Nonlinear differential dynamics of Gaussian States

Julio A. López-Saldívar, Margarita A. Man’ko, Vladimir I. Man’ko

The dynamics of Gaussian states for systems with Hamiltonians, quadratic in the position and momentum operators, gives rise to the definition of a system of nonlinear coupled differential equations for the density matrix parameters of the system states. In this work, we show that, using the Gaussian-state covariance-matrix evolution, it is possible to solve the system of these nonlinear equations. Some examples of applications of the method developed are given for one- and two-dimensional quadratic-system Gaussian states. It is noted that this formalism can also be used to find new constants of motion related to the covariance matrices of quadratic systems.

AIP conference proceedings•Jan 1, 2021•2 citations

Prediction and retrodiction with continuously monitored Gaussian states

Jinglei Zhang, Klaus Mølmer

Gaussian states of quantum oscillators are fully characterized by the mean values and the covariance matrix of their quadrature observables. We consider the dynamics of a system of oscillators subject to interactions, damping, and continuous probing which maintain their Gaussian state property. Such dynamics is found in many physical systems that can therefore be efficiently described by the ensuing effective representation of the density matrix $\ensuremath{\rho}(t)$. Our probabilistic knowledge about the outcome of measurements on a quantum system at time $t$ is not only governed by $\ensuremath{\rho}(t)$ conditioned on the evolution and measurement outcomes obtained until time $t$ but is also modified by any information acquired after $t$. It was shown [S. Gammelmark, B. Julsgaard, and K. M\o{}lmer, Phys. Rev. Lett. 111, 160401 (2013)] that this information is represented by a supplementary matrix, $E(t)$. We show here that the restriction of the dynamics of $\ensuremath{\rho}(t)$ to Gaussian states implies that the matrix $E(t)$ is also fully characterized by a vector of mean values and a covariance matrix. We derive the dynamical equations for these quantities and we illustrate their use in the retrodiction of measurements on Gaussian systems.

arXiv:1710.04950•Physical Review A•Dec 26, 2017•45 citations

Continuous Variable Systems

Claude Fabre, Rodrigo G. Cortiñas

Theoretical chapter devoted to the detailed description of continuous variable (CV) systems by consideringthe "phase space," that is spanned by position and momentum for massive particles, quadratures for a quantum electromagnetic field, and phase and charge for electrical circuits. It introduces tools like the Glauber, Husimi, or Dirac phase–space functions, and in more details the Wigner function, that are convenient to describe CV quantum states and their time evolution using the Moyal equation. The chapter gives examples of Wigner functions and their time evolution in the presence of dissipation. It then defines symplectic quantum maps that are simple and important cases of Hamiltonian evolution and are simply related to the covariance matrix containingvariances and correlations. It details the characterization of the quantum processes using the Williamson reduction and Bloch–Messiah decomposition. It discusses Gaussian and non-Gaussian states and the specific measurement procedures for CV states, such as homodyne and double homodyne detection. It introduces the EPR entangled state and, finally, describes how to characterize entanglement and unconditionally teleport Gaussian quantum states.

Cambridge University Press eBooks•Jul 27, 2023•1 citation

Generation and evolution of entanglement in open quantum dynamics

Аурелиан Исар

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment was given. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state was described. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. The time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state was analyzed.

Optics and Spectroscopy•Feb 1, 2010•0 citations
The covariance matrix contains the complete information about the second-order expectation values of the mode quadratures (position and momentum operators) of the system. Due to its prominence in studies of continuous variable systems, most significantly Gaussian states, special emphasis is put on time evolution models that result in self-contained equations for the covariance matrix. So far, despite not being explicitly implied by this requirement, virtually all such models assume a so-called quadratic, or second-order case, in which the generator of the evolution is at most second-order in the mode quadratures. Here, we provide an explicit model of covariance matrix evolution of infinite order. Furthermore, we derive the solution, including stationary states, for a large subclass of proposed evolutions. Our findings challenge the contemporary understanding of covariance matrix dynamics and may give rise to new methods and improvements in quantum technologies employing continuous variable systems.
In the 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 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 Schrödinger 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.
Covariance matrices lie at the heart of the powerful and well-established symplectic framework for describing continuous-variable Gaussian quantum states. However, since this framework only relies on first- and second-order moments, it is not sufficient for the analysis of non-Gaussian states because their higher-order moments are essential to capture some of their key properties. Here, we define higher-order covariance matrices -- more precisely, covariance matrices built from higher-order quadrature monomials -- which provide a simple way to evaluate the effect of Gaussian transformations on non-Gaussian states and can be used, for example, to address nonlinear squeezing or non-Gaussian nullifiers. Higher-order covariance matrices can be estimated from homodyne measurement data using only a limited number of quadrature angles, which involves matrices of moderate dimension compared with a full simulation in the Fock basis. The dimension of a higher-order covariance matrix does not depend on the span of the quantum states in Fock basis and, furthermore, scales only polynomially with the number of modes.
The dynamics of Gaussian states for systems with Hamiltonians, quadratic in the position and momentum operators, gives rise to the definition of a system of nonlinear coupled differential equations for the density matrix parameters of the system states. In this work, we show that, using the Gaussian-state covariance-matrix evolution, it is possible to solve the system of these nonlinear equations. Some examples of applications of the method developed are given for one- and two-dimensional quadratic-system Gaussian states. It is noted that this formalism can also be used to find new constants of motion related to the covariance matrices of quadratic systems.
Gaussian states of quantum oscillators are fully characterized by the mean values and the covariance matrix of their quadrature observables. We consider the dynamics of a system of oscillators subject to interactions, damping, and continuous probing which maintain their Gaussian state property. Such dynamics is found in many physical systems that can therefore be efficiently described by the ensuing effective representation of the density matrix $\ensuremath{\rho}(t)$. Our probabilistic knowledge about the outcome of measurements on a quantum system at time ttt is not only governed by $\ensuremath{\rho}(t)$ conditioned on the evolution and measurement outcomes obtained until time ttt but is also modified by any information acquired after ttt. It was shown [S. Gammelmark, B. Julsgaard, and K. M\o{}lmer, Phys. Rev. Lett. 111, 160401 (2013)] that this information is represented by a supplementary matrix, E(t)E(t)E(t). We show here that the restriction of the dynamics of $\ensuremath{\rho}(t)$ to Gaussian states implies that the matrix E(t)E(t)E(t) is also fully characterized by a vector of mean values and a covariance matrix. We derive the dynamical equations for these quantities and we illustrate their use in the retrodiction of measurements on Gaussian systems.
Theoretical chapter devoted to the detailed description of continuous variable (CV) systems by consideringthe "phase space," that is spanned by position and momentum for massive particles, quadratures for a quantum electromagnetic field, and phase and charge for electrical circuits. It introduces tools like the Glauber, Husimi, or Dirac phase–space functions, and in more details the Wigner function, that are convenient to describe CV quantum states and their time evolution using the Moyal equation. The chapter gives examples of Wigner functions and their time evolution in the presence of dissipation. It then defines symplectic quantum maps that are simple and important cases of Hamiltonian evolution and are simply related to the covariance matrix containingvariances and correlations. It details the characterization of the quantum processes using the Williamson reduction and Bloch–Messiah decomposition. It discusses Gaussian and non-Gaussian states and the specific measurement procedures for CV states, such as homodyne and double homodyne detection. It introduces the EPR entangled state and, finally, describes how to characterize entanglement and unconditionally teleport Gaussian quantum states.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment was given. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state was described. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. The time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state was analyzed.