The statistical error in any estimation can be reduced by repeating the measurement and averaging the results. The central limit theorem implies that the reduction is proportional to the square root of the number of repetitions. Quantum metrology is the use of quantum techniques such as entanglement to yield higher statistical precision than purely classical approaches. In this Review, we analyse some of the most promising recent developments of this research field and point out some of the new experiments. We then look at one of the major new trends of the field: analyses of the effects of noise and experimental imperfections.

Advances in quantum metrology

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalities, entanglement witnesses, the determination of nonlinear properties of a quantum state via measurements on several copies, and spin squeezing inequalities. An emphasis is given on the theory and application of entanglement witnesses. We also discuss several experiments, where some of the presented methods have been implemented.

Entanglement detection

Annals of physics

Tables of integrals

https://cds.cern.ch/record/352811/files/CM-P00058746.pdf

Exact integral equation for pion-pion scattering involving only physical region partial waves

We show that when a suitable entanglement-generating unitary operator depending on a parameter is applied on $N$ qubits in parallel, a precision of the order of ${2}^{\ensuremath{-}N}$ in estimating the parameter may be achieved. This exponentially improves the precision achievable in classical and in quantum nonentangling strategies.

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Exponentially Enhanced Quantum Metrology

For multiparticle systems we formulate the notions of signal locality (i.e., absence of faster-than-light signaling) and Einstein-Bell locality (or local realism) and obtain inequalities between experimental correlation functions to test them. Quantum theory obeys signal locality but violates Einstein-Bell locality by a factor ${2}^{(\mathit{n}\mathrm{\ensuremath{-}}1)/2}$ for n-particle systems.

Tests of signal locality and Einstein-Bell locality for multiparticle systems

We derive from a dynamical symmetry property that the linear and nonlinear Schr\"odinger equations with harmonic potential possess an infinite string of shape-preserving coherent wave-packet states with classical motion. Unlike the Schr\"odinger state with $\ensuremath{\Delta}x\ensuremath{\Delta}p=\frac{\ensuremath{\hbar}}{2}$, the uncertainty product can be arbitrarily large for these states showing that classical motion is not necessarily linked with minimum uncertainty. We obtain a generalization of Sudarshan's diagonal coherent-state representation in terms of these states.

S. M. Roy

Papers

Exact integral equation for pion-pion scattering involving only physical region partial waves

Exponentially Enhanced Quantum Metrology

Tests of signal locality and Einstein-Bell locality for multiparticle systems

Generalized coherent states and the uncertainty principle

Semi-relativistic stability and critical mass of a system of spinless bosons in gravitational interaction