From particle counting to Gaussian tomography

Parthasarathy, K. R.; Sengupta, Ritabrata

doi:10.1142/s021902571550023x

Cited by 23 publications

(26 citation statements)

References 20 publications

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“…Since the CCR matrix Θ is assumed to be nonsingular, (14) is equivalent to det(M T JM) = 0 (25) in view of the parameterisation of the matrix B in (6) in terms of the coupling matrix M. A necessary condition for (25) is that M is of full column rank: M T M ≻ 0, and hence, m n (that is, the number of oscillator modes does not exceed the number of external field channels). Theorem 1 characterizes the kernel Λ in (12) as the Green function for the BVP (15), (16).…”

Section: Integral Operator With the Commutator Kernelmentioning

confidence: 99%

“…In addition to their relevance to quantum risk-sensitive control, these results can contribute to the study of operator exponential structures which are being actively researched in mathematical physics and quantum probability (for example, in the context of operator algebras [1], moment-generating functions for quadratic Hamiltonians [25] and the quantum Lévy area [3], [11]).…”

Section: Introductionmentioning

confidence: 99%

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A Quantum Karhunen-Loeve Expansion and Quadratic-Exponential Functionals for Linear Quantum Stochastic Systems

Vladimirov

Petersen

James

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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This paper is concerned with multimode open quantum harmonic oscillators and quadratic-exponential functionals (QEFs) as quantum risk-sensitive performance criteria. Such systems are described by linear quantum stochastic differential equations driven by multichannel bosonic fields. We develop a finite-horizon expansion for the system variables using the eigenbasis of their two-point commutator kernel with noncommuting positionmomentum pairs as coefficients. This quantum Karhunen-Loeve expansion is used in order to obtain a Girsanov type representation for the quadratic-exponential functions of the system variables. This representation is valid regardless of a particular system-field state and employs the averaging over an auxiliary classical Gaussian random process whose covariance operator is defined in terms of the quantum commutator kernel. We use this representation in order to relate the QEF to the moment-generating functional of the system variables. This result is also specified for the invariant multipoint Gaussian quantum state when the oscillator is driven by vacuum fields.

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Section: Integral Operator With the Commutator Kernelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Quantum Karhunen-Loeve Expansion and Quadratic-Exponential Functionals for Linear Quantum Stochastic Systems

Vladimirov

Petersen

James

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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“…(11) has a Gaussian Wigner function. Since displacement operations as well as all experimental noise and losses are represented by Gaussian channels [29], the state always remains Gaussian and can easily be fully characterised; see for example [30]. One can then use this experimentally characterized state in Eq.…”

Section: Effect Of Imperfectionsmentioning

confidence: 99%

Optomechanical state reconstruction and nonclassicality verification beyond the resolved-sideband regime

Shähandeh

Ringbauer

2019

Quantum

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Quantum optomechanics uses optical means to generate and manipulate quantum states of motion of mechanical resonators. This provides an intriguing platform for the study of fundamental physics and the development of novel quantum devices.Yet, the challenge of reconstructing and verifying the quantum state of mechanical systems has remained a major roadblock in the field. Here, we present a novel approach that allows for tomographic reconstruction of the quantum state of a mechanical system without the need for extremely high quality optical cavities. We show that, without relying on the usual state transfer presumption between light an mechanics, the full optomechanical Hamiltonian can be exploited to imprint mechanical tomograms on a strong optical coherent pulse, which can then be read out using well-established techniques. Furthermore, with only a small number of measurements, our method can be used to witness nonclassical features of mechanical systems without requiring full tomography. By relaxing the experimental requirements, our technique thus opens a feasible route towards verifying the quantum state of mechanical resonators and their nonclassical behaviour in a wide range of optomechanical systems.

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“…Substitution of (36) into (33) leads to (29). From (29), it follows that if the initial state ϖ of the system is Gaussian [48,50] (that is, ln Φ(0, u) is a quadratic function of u ∈ R n ), then so is its reduced quantum state at subsequent moments of time t > 0. Furthermore, since the matrix A is Hurwitz, the relations (28), (29) and the continuity lim u→0 Φ(s, u) = Φ(s, 0) = 1 imply the pointwise convergence of the QCF:…”

Section: Linear Quantum Stochastic Systemsmentioning

confidence: 99%

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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From particle counting to Gaussian tomography

Cited by 23 publications

References 20 publications

A Quantum Karhunen-Loeve Expansion and Quadratic-Exponential Functionals for Linear Quantum Stochastic Systems

A Quantum Karhunen-Loeve Expansion and Quadratic-Exponential Functionals for Linear Quantum Stochastic Systems

Optomechanical state reconstruction and nonclassicality verification beyond the resolved-sideband regime

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

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