Quantum stochastic calculus

Холево, Александр Семенович

doi:10.1007/bf01095973

Cited by 37 publications

(44 citation statements)

References 18 publications

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“…Proof The factorization (21) can be obtained by repeatedly applying the Weyl CCRs (13), from which it follows by induction that…”

Section: Lemmamentioning

confidence: 99%

“…is the kth standard basis vector in R n , so that W u k e k = e iu k X k , and (22) establishes (21). Here, use is made of (12) and the identity ∑ 1 j<k n θ jk u j u k = 1 2 u T Θ ⋄ u, where, in view of (20), the real symmetric matrix Θ ⋄ inherits the upper off-diagonal entries and the zero main diagonal from Θ .…”

Section: Lemmamentioning

confidence: 99%

“…Such systems include quantum harmonic oscillators (QHOs) and their open versions (OQHOs) [16] interacting with the environment, which can involve quantum fields (for example, nonclassical light), other quantum systems or classical measuring devices, controllers and actuators. These systems and their interconnections are usually described in terms of linear quantum stochastic differential equations (QSDEs) and other tools of the Hudson-Parthasarathy quantum stochastic calculus [21,25,40,43]. In combination with quantum feedback network theory [19,30], this approach provides an important modelling paradigm in linear quantum systems theory [44,56,58].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Proof The factorization (21) can be obtained by repeatedly applying the Weyl CCRs (13), from which it follows by induction that…”

Section: Lemmamentioning

confidence: 99%

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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“…. ,W m , which are self-adjoint operators on a boson Fock space [18,27], modelling the external fields with a positive semi-definite Itô matrix…”

Section: Linear Quantum Stochastic Systemsmentioning

confidence: 99%

Quantum linear coherent controller synthesis: A linear fractional representation approach

Sichani

Petersen

2017

Automatica

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This paper is concerned with a linear fractional representation approach to the synthesis of linear coherent quantum controllers for a given linear quantum plant. The plant and controller represent open quantum harmonic oscillators and are modelled by linear quantum stochastic differential equations. The feedback interconnections between the plant and the controller are assumed to be established through quantum bosonic fields. In this framework, conditions for the stabilization of a given linear quantum plant via linear coherent quantum feedback are addressed using a stable factorization approach. The class of all stabilizing quantum controllers is parameterized in the frequency domain. Coherent quantum weighted H 2 and H ∞ control problems for linear quantum systems are formulated in the frequency domain. Finally, a projected gradient descent scheme is outlined for the coherent quantum weighted H 2 control problem.

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“…These developments (see, for example, [23,31,33,52,53]) are particularly focused on open quantum systems whose internal dynamics are affected by interaction with the environment [8]. In such systems, the evolution of dynamic variables (as noncommutative operators on a Hilbert space) is often modelled using the Hudson-Parthasarathy calculus [18,21,35] which provides a rigorous framework of quantum stochastic differential equations (QSDEs) driven by quantum Wiener processes on symmetric Fock spaces. In particular, linear QS-DEs model open quantum harmonic oscillators (OQHOs) [13] whose dynamic variables (such as the position and momentum or annihilation and creation operators [29,42]) satisfy canonical commutation relations (CCRs).…”

Section: Introductionmentioning

confidence: 99%

Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action

Vladimirov¹,

Petersen²

2016

2016 IEEE Conference on Norbert Wiener in the 21st Century (21CW)

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This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of second-order moments of the system variables. Small-gain-theorem bounds are obtained for the back-action of the observer on the covariance dynamics of the plant in terms of the plant-observer coupling. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant-observer coupling matrix in order to mitigate the back-action effect. For the discounted mean square optimal CQF problem with penalized back-action, we establish first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic techniques, this set of equations is represented in a more explicit form for equally dimensioned plant and observer. For a class of such observers with autonomous estimation error dynamics, we obtain a solution of the CQF problem and outline a homotopy method. The computation of the performance criteria and the observer synthesis are illustrated by numerical examples.

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Quantum stochastic calculus

Cited by 37 publications

References 18 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

Quantum linear coherent controller synthesis: A linear fractional representation approach

Directly coupled observers for quantum harmonic oscillators with discounted mean square cost functionals and penalized back-action

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