A Schwinger disentangling theorem

Cross, Daniel J.; Gilmore, Robert

doi:10.1063/1.3501027

Cited by 4 publications

(7 citation statements)

References 4 publications

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“…The following lemma is similar to the commutation relations for bilinear functions of annihilation and creation operators (see, for example, [26,Appendix B] and [45,Lemma 4.2]), whose exponentials are considered in Schwinger's theorems [8].…”

Section: A Commutator For Quadratic Forms Of Quantum Variables Satisfmentioning

confidence: 99%

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Parametric randomization, complex symplectic factorizations, and quadratic-exponential functionals for Gaussian quantum states

Vladimirov

Petersen

James

2019

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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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 (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive filtering and control problems for linear quantum stochastic systems. We employ a Lie-algebraic correspondence between complex symplectic matrices and quadratic-exponential functions of system variables of a quantum harmonic oscillator. The complex symplectic factorizations are used together with a parametric randomization of the quasi-characteristic or moment-generating functions according to an auxiliary classical Gaussian distribution. This reduces the QEF to an exponential moment of a quadratic form of classical Gaussian random variables with a complex symmetric matrix and is applicable to recursive computation of such moments.

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Section: A Commutator For Quadratic Forms Of Quantum Variables Satisfmentioning

confidence: 99%

“…There is an analogy between (C6) (and its application to (C9)) and the factorizations in [8,Eqs. (5), (6)].…”

Section: A Commutator For Quadratic Forms Of Quantum Variables Satisfmentioning

confidence: 99%

Parametric randomization, complex symplectic factorizations, and quadratic-exponential functionals for Gaussian quantum states

Vladimirov

Petersen

James

2019

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…Lemma 1 Suppose the OQHO, governed by the linear QSDE (9), is driven by the input fields in the vacuum state. Then the QCF Φ for the system variables in (26) satisfies a linear functional equation…”

Section: Linear Quantum Stochastic Systemsmentioning

confidence: 99%

“…which gives rise to the field-related term 2Θ M T JM = − 1 2 BJB T Θ −1 in the matrix A in (13). The other part i[H, X] = 2Θ RX of the drift vector AX in the QSDE (9) comes from the quadratic nature of the Hamiltonian H in (8) in combination with the CCRs (5) and describes the internal dynamics of the system variables which they would have if the system were isolated from the environment. Also, the representation of the dispersion matrix B = −i[X, L T ] of the QSDE (9) in (13) follows from the linear dependence of the coupling operators in (8) on the system variables and the CCRs (5).…”

Section: Linear Quantum Stochastic Systemsmentioning

confidence: 99%

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Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

2018

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This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. An integro-differential equation is obtained for the time evolution of this quadratic-exponential functional, which is compared with the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaussian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite-horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. We also outline further approximations of the cost functional, based on higher-order cumulants and their growth rates, together with large deviations estimates. For comparison, an auxiliary classical Gaussian Markov diffusion process is considered in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different higher-order moments relevant to the risk-sensitive criteria. The results of the paper are also demonstrated by a numerical example and may find applications to coherent quantum risk-sensitive control problems, where the plant and controller form a fully quantum closed-loop system, and other settings with nonquadratic cost functionals.Keywords Linear quantum stochastic system · Gaussian quantum state · Risk-sensitive quantum control Mathematics Subject Classification (2010) 81S25 · 81S05 · 81S22 · 81P16 · 81P40 · 81Q93 · 81Q10 · 60G15

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Method of Generating N-dimensional Isochronous Nonsingular Hamiltonian Systems

Devi¹,

Pradeep²,

Chandrasekar³

et al. 2021

JNMP

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In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that a class of singular Hamiltonian systems obtained through the Ω-modified procedure is equivalent to constrained Newtonian systems. Even though such systems admit isochronous oscillations, they are effectively one degree of freedom systems due to the constraints. Then we generalize the procedure in terms of Ω i -modified Hamiltonians and identify suitable canonically conjugate coordinates such that the constructed Ω i -modified Hamiltonian is nonsingular and the corresponding Newton's equation of motion is constraint free. The procedure is first illustrated for two dimensional systems and subsequently extended to N-dimensional systems. The general solution of these systems are obtained by integrating the underlying equations and is shown to admit isochronous as well as amplitude independent quasiperiodic solutions depending on the choice of parameters.

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A Schwinger disentangling theorem

Cited by 4 publications

References 4 publications

Parametric randomization, complex symplectic factorizations, and quadratic-exponential functionals for Gaussian quantum states

Parametric randomization, complex symplectic factorizations, and quadratic-exponential functionals for Gaussian quantum states

Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

Method of Generating N-dimensional Isochronous Nonsingular Hamiltonian Systems

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