This paper is concerned with integro-differential identities which are known in statistical signal processing as Price's theorem for expectations of nonlinear functions of jointly Gaussian random variables. We revisit these relations for classical variables by using the Frechet differentiation with respect to covariance matrices, and then show that Price's theorem carries over to a quantum mechanical setting. The quantum counterpart of the theorem is established for Gaussian quantum states in the framework of the Weyl functional calculus for quantum variables satisfying the Heisenberg canonical commutation relations. The quantum mechanical version of Price's theorem relates the Frechet derivative of the generalized moment of such variables with respect to the real part of their quantum covariance matrix with other moments. As an illustrative example, we consider these relations for quadratic-exponential moments which are relevant to risk-sensitive quantum control.This paper is concerned with integro-differential identities which are known in statistical signal processing as Price's theorem for expectations of nonlinear functions of jointly Gaussian random variables. We revisit these relations for classical variables by using the Frechet differentiation with respect to covariance matrices, and then show that Price's theorem carries over to a quantum mechanical setting. The quantum counterpart of the theorem is established for Gaussian quantum states in the framework of the Weyl functional calculus for quantum variables satisfying the Heisenberg canonical commutation relations. The quantum mechanical version of Price's theorem relates the Frechet derivative of the generalized moment of such variables with respect to the real part of their quantum covariance matrix with other moments. As an illustrative example, we consider these relations for quadratic-exponential moments which are relevant to risk-sensitive quantum control.

arXiv:1409.4377Australian Control ConferenceNov 1, 20144 citations

Hermite–Gaussian model for quantum states

Marcelo Losada, Ignacio S. Gomez, Federico Holik

arXiv:1811.01207Physica A Statistical Mechanics and its ApplicationsJun 19, 20191 citation

Squeezing versus photon-number fluctuations

Giacomo Mauro D’Ariano, S. Morosi, Mario Rasetti +2 more

After deriving a general formula for the quantum probability distribution function higher moments, we apply it to the multiphoton squeezed states [the usual Gaussian and the new non-Gaussian Weyl-Heisenberg, SU(2), and SU(1,1)]. The resulting moments are discussed as functions of the photon-number fluctuations. General criteria are considered to determine optimal squeezing properties with respect to photon-number noise. There result interesting generalized uncertainty relations in the form of scaling laws.After deriving a general formula for the quantum probability distribution function higher moments, we apply it to the multiphoton squeezed states [the usual Gaussian and the new non-Gaussian Weyl-Heisenberg, SU(2), and SU(1,1)]. The resulting moments are discussed as functions of the photon-number fluctuations. General criteria are considered to determine optimal squeezing properties with respect to photon-number noise. There result interesting generalized uncertainty relations in the form of scaling laws.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsOct 15, 198736 citations

Study on quantum properties of non gaussian states

Priyanka Chowdhury

Jan 1, 20160 citations

Gaussian States in Quantum Information

Alessandro Ferraro, Stefano Olivares, Matteo G. A. Paris

Jan 1, 2005291 citations

Evolution of quasi-characteristic functions in quantum stochastic systems with Weyl quantization of energy operators

Igor G. Vladimirov

This paper considers open quantum systems whose dynamic variables satisfy canonical commutation relations and are governed by Markovian Hudson-Parthasarathy quantum stochastic differential equations driven by external bosonic fields. The dependence of the Hamiltonian and the system-field coupling operators on the system variables is represented using the Weyl functional calculus. This leads to an integro-differential equation (IDE) for the evolution of the quasi-characteristic function (QCF) which encodes the dynamics of mixed moments of the system variables. Unlike quantum master equations for reduced density operators, this IDE involves only complex-valued functions on finite-dimensional Euclidean spaces and extends the Wigner-Moyal phase-space approach for quantum stochastic systems. The dynamics of the QCF and the related Wigner quasi-probability density function (QPDF) are discussed in more detail for the case when the coupling operators depend linearly on the system variables and the Hamiltonian has a nonquadratic part represented in the Weyl quantization form. For this class of quantum stochastic systems, we also consider an approximate computation of invariant states and discuss the deviation from Gaussian quantum states in terms of the $χ^2$ -divergence (or the second-order Renyi relative entropy) applied to the QPDF. The results of the paper may find applications to investigating different aspects of the moment stability, relaxation dynamics and invariant states in open quantum systems.This paper considers open quantum systems whose dynamic variables satisfy canonical commutation relations and are governed by Markovian Hudson-Parthasarathy quantum stochastic differential equations driven by external bosonic fields. The dependence of the Hamiltonian and the system-field coupling operators on the system variables is represented using the Weyl functional calculus. This leads to an integro-differential equation (IDE) for the evolution of the quasi-characteristic function (QCF) which encodes the dynamics of mixed moments of the system variables. Unlike quantum master equations for reduced density operators, this IDE involves only complex-valued functions on finite-dimensional Euclidean spaces and extends the Wigner-Moyal phase-space approach for quantum stochastic systems. The dynamics of the QCF and the related Wigner quasi-probability density function (QPDF) are discussed in more detail for the case when the coupling operators depend linearly on the system variables and the Hamiltonian has a nonquadratic part represented in the Weyl quantization form. For this class of quantum stochastic systems, we also consider an approximate computation of invariant states and discuss the deviation from Gaussian quantum states in terms of the $χ^2$-divergence (or the second-order Renyi relative entropy) applied to the QPDF. The results of the paper may find applications to investigating different aspects of the moment stability, relaxation dynamics and invariant states in open quantum systems.

arXiv:1512.08751Dec 29, 20156 citations

χ^2

-divergence (or the second-order Renyi relative entropy) applied to the QPDF. The results of the paper may find applications to investigating different aspects of the moment stability, relaxation dynamics and invariant states in open quantum systems.