Information Geometry of Complex Hamiltonians and Exceptional Points

Brody, Dorje C.; Graefe, Eva-Maria

doi:10.3390/e15093361

Cited by 44 publications

(42 citation statements)

References 68 publications

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“…For example, the parameter can appear with different orders in the eigenvalues of the Hamiltonian, or even in the eigenstates of the Hamiltonian. An understanding of the quantum limits in estimating this kind of general parameter is emerging (e.g., [54] from the view of information geometry), but is still rather limited so far, which restricts the potential range of applications of quantum mechanics to metrology. This paper extends quantum metrology to estimating a general parameter of a Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Quantum metrology for a general Hamiltonian parameter

2014

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Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-1 2 particle and compare the results for estimating the amplitude and the direction of the magnetic field.

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Section: Introductionmentioning

confidence: 99%

Quantum metrology for a general Hamiltonian parameter

2014

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“…More general Hamiltonians have recently started to attract attention 202 , since they permit the application of quantum metrology to a more general class of problems such as timevarying fields 203,204 and in gradient magnetometry 205 . It is well known that a pure state parameterised by a multiplicative Hamiltonian of the form H j = θ j G j for a time t, the is given by 22,112…”

Section: B Hamiltonians With Non-multiplicative Factorsmentioning

confidence: 99%

Geometric perspective on quantum parameter estimation

Sidhu

Kok

2020

AVS Quantum Science

176

132

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Quantum metrology holds the promise of an early practical application of quantum technologies, in which measurements of physical quantities can be made with much greater precision than what is achievable with classical technologies. In this review, we collect some of the key theoretical results in quantum parameter estimation by presenting the theory for the quantum estimation of a single parameter, multiple parameters, and optical estimation using Gaussian states. We give an overview of results in areas of current research interest, such as Bayesian quantum estimation, noisy quantum metrology, and distributed quantum sensing. We address the question how minimum measurement errors can be achieved using entanglement as well as more general quantum states. This review is presented from a geometric perspective. This has the advantage that it unifies a wide variety of estimation procedures and strategies, thus providing a more intuitive big picture of quantum parameter estimation. CONTENTS

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“…Standard perturbation theory relies on Taylor expansions of differentiable functions while, near EPs, eigenvalues change non-analytically in response to small matrix perturbations. Instead, one needs to use a Jordan-form-based perturbative expansion [39,40]. Although Jordan-vector perturbation theory is well known in linear algebra, along with related results on resolvent operators, these algebraic facts have not previously been applied to analyze Purcell/Petermann enhancement or LDOS in a general EP setting.…”

Section: Introductionmentioning

confidence: 99%

General theory of spontaneous emission near exceptional points

Pick¹,

Zhen²,

et al. 2017

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Abstract:We present a general theory of spontaneous emission at exceptional points (EPs)-exotic degeneracies in non-Hermitian systems. Our theory extends beyond spontaneous emission to any light-matter interaction described by the local density of states (e.g., absorption, thermal emission, and nonlinear frequency conversion). Whereas traditional spontaneous-emission theories imply infinite enhancement factors at EPs, we derive finite bounds on the enhancement, proving maximum enhancement of 4 in passive systems with second-order EPs and significantly larger enhancements (exceeding 400×) in gain-aided and higher-order EP systems. In contrast to non-degenerate resonances, which are typically associated with Lorentzian emission curves in systems with low losses, EPs are associated with non-Lorentzian lineshapes, leading to enhancements that scale nonlinearly with the resonance quality factor. Our theory can be applied to dispersive media, with proper normalization of the resonant modes. References and links1. E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681--681 (1946). (CRC Press, 1995, vol. X). 3. S. V. Gaponenko, Introduction to Nanophotonics (Cambridge University, 2010 H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities

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Information Geometry of Complex Hamiltonians and Exceptional Points

Cited by 44 publications

References 68 publications

Quantum metrology for a general Hamiltonian parameter

Quantum metrology for a general Hamiltonian parameter

Geometric perspective on quantum parameter estimation

General theory of spontaneous emission near exceptional points

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