Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance

Braunstein, Samuel L.; Caves, Carlton M.; Milburn, G. J.

doi:10.1006/aphy.1996.0040

Cited by 671 publications

(832 citation statements)

References 30 publications

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“…In its simplest version a typical quantum estimation problem [8][9][10][11][12][13][14] consists in recovering the value of a continuous parameter x (say the phase ϕ of Fig. 1) which is encoded into a fixed set of states ρ x of a quantum system S. As in the example of Fig.…”

Section: Basics On Quantum Estimation For Statesmentioning

confidence: 99%

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Advances in quantum metrology

2011

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In classical estimation theory, the central limit theorem implies that the statistical error in a measurement outcome can be reduced by an amount proportional to n −1/2 by repeating the measures n times and then averaging. Using quantum effects, such as entanglement, it is often possible to do better, decreasing the error by an amount proportional to n −1 . Quantum metrology is the study of those quantum techniques that allow one to gain advantages over purely classical approaches.In this review, we analyze some of the most promising recent developments in this research field. Specifically, we deal with the developments of the theory and point out some of the new experiments. Then we look at one of the main new trends of the field, the analysis of how the theory must take into account the presence of noise and experimental imperfections.Any measurement consists in three parts: the preparation of a probe, its interaction with the system to be measured, and the probe readout. This process is often plagued by statistical or systematic errors. The source of the former can be accidental (e.g. deriving from an insufficient control of the probes or of the measured system) or fundamental (e.g. deriving from the Heisenberg uncertainty relations). Whatever their origin, we can reduce their effect by repeating the measurement and averaging the resulting outcomes. This is a consequence of the central-limit theorem: given a large number n of independent measurement results (each having a standard deviation ∆σ), their average will converge to a Gaussian distribution with standard deviation ∆σ/ √ n, so that the error scales as n −1/2 . In quantum mechanics this behavior is referred to as "standard quantum limit" (SQL) and is associated with procedures which do not fully exploit the quantum nature of the system under investigation 1 . Notably it is possible to do better when one employs quantum effects, such as entanglement among the probing devices employed for the measurements, e.g. see Refs. [1][2][3][4][5][6]. Consequently, the SQL is not a fundamental quantum mechanical bound as it can be surpassed by using "non-classical" strategies. Nonetheless, through Heisenberg-like uncertainty relations, quantum mechanics still sets ultimate limits in precision which are typically referred to as "Heisenberg bounds". In Fig. 1 we present a simple example which may be useful to un-1 In quantum optics, the n −1/2 scaling is also indicated as "shot noise", since it is connected to the discrete nature of the radiation that can be heard as "shots" in a photon counter operating in Geiger mode.derstand the quantum enhancement. Part of the emerging field of quantum technology [7], quantum metrology studies these bounds and the (quantum) strategies which allows us to attain them. More generally it deals with measurement and discrimination procedures that receive some kind of enhancement (in precision, efficiency, simplicity of implementation, etc.) through the use of quantum effects. This paper aims to review some of the most recent developmen...

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Section: Basics On Quantum Estimation For Statesmentioning

confidence: 99%

“…(1) with respect to all possible POVMs E (n) one then gets the inequality [8][9][10][11][12][13][14] …”

Section: Basics On Quantum Estimation For Statesmentioning

confidence: 99%

Advances in quantum metrology

2011

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“…It quantifies the information that one can extract about a parameter from the observed probability distribution. Moving into the quantum regime, the extension of Fisher information is known as the quantum Fisher information (QFI), which predicts the theoretical achievable limit on the measurement precision in quantum metrology [2,3,4,5]. Besides the application in quantum metrology, QFI also plays an important role in the field of quantum information theory.…”

Section: Introductionmentioning

confidence: 99%

“…According to quantum Cramér-Rao theorem [2,3,4,5], the accuracy of estimation is asymptotically bounded by the inverse of QFI. Thus, the quantity χ 2 in equation (1) happens to be the average parameter estimation precision (APEP) of single particles.…”

Section: Introductionmentioning

confidence: 99%

Quantum Fisher information and spin squeezing in one-axis twisting model

Zhong¹,

Liu²,

Ma³

et al. 2014

Chinese Phys. B

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Abstract. We consider the quantum Fisher information and spin squeezing in oneaxis twisting model with a coherent spin state |θ 0 , φ 0 . We analytically discuss the dependence of the two parameters: spin squeezing parameter ξ 2 K and the average parameter estimation precision χ 2 on the polar angle θ 0 and the azimuth angle φ 0 . Moreover, we discuss the effects of the collisional dephasing on the dynamics of the two parameters. In this case, the analytical solution of ξ 2 K is also obtained.

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“…The smallest value of d that can be measured is when this overlap is nearly zero, which corresponds to d min ∼ 4σ/ √ N . This result for a minimum resolvable position can be compared with the quantum Fisher information [14] in the state | (g) about the parameter g. The quantum Fisher information for pure states is defined as…”

Section: Spatial Shift Of Independent Meter Statesmentioning

confidence: 99%

Heisenberg scaling with weak measurement: a quantum state discrimination point of view

Jordan¹,

Tollaksen²,

Troupe³

et al. 2015

Quantum Stud.: Math. Found.

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We examine the results of the paper "Precision metrology using weak measurements" (Zhang et al. arXiv:1310.5302, 2013) from a quantum state discrimination point of view. The Heisenberg scaling of the photon number for the precision of the interaction parameter between coherent light and a spin one-half particle (or pseudospin) has a simple interpretation in terms of the interaction rotating the quantum state to an orthogonal one. To achieve this scaling, the information must be extracted from the spin rather than from the coherent state of light, limiting the applications of the method to phenomena such as cross-phase modulation. We next investigate the effect of dephasing noise and show a rapid degradation of precision, in agreement with general results in the literature concerning Heisenberg scaling metrology. We also demonstrate that a von Neumann-type measurement interaction can display a similar effect with no system/meter entanglement.

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Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance

Cited by 671 publications

References 30 publications

Advances in quantum metrology

Advances in quantum metrology

Quantum Fisher information and spin squeezing in one-axis twisting model

Heisenberg scaling with weak measurement: a quantum state discrimination point of view

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