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arXiv:0711.4840 · Physical Review Letters

Entanglement, Nonlinear Dynamics, and the Heisenberg Limit

Luca Pezzè, Augusto Smerzi

Published Mar 10, 2009• 874 citations
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Abstract

We show that quantum Fisher information provides a sufficient condition to recognize multiparticle entanglement in an N qubit state. The same criterion gives a necessary and sufficient condition for sub-shot-noise phase sensitivity in the estimation of a collective rotation angle theta. The analysis therefore singles out the class of entangled states which are useful to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the nonlinear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.

DOI
10.1103/physrevlett.102.100401

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We show that quantum Fisher information provides a sufficient condition to recognize multiparticle entanglement in an N qubit state. The same criterion gives a necessary and sufficient condition for sub-shot-noise phase sensitivity in the estimation of a collective rotation angle theta. The analysis therefore singles out the class of entangled states which are useful to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the nonlinear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.
In terms of quantum Fisher information, a quantity χ2\chi^{2}χ2 was introduced by Pezz\'{e} and Smerzi, which is a multiparticle entanglement measure, and provides a necessary and sufficient condition for sub-shot-noise phase estimation sensitivity. We derive a general expression of χ2\chi ^{2}χ2 for arbitrary symmetric multiqubit states with nonzero mean spins. It is shown that the entangled symmetric states are useful for phase sensitivity beyond the shot-noise limit. Using the expression, we explicitly examine a series of superpositions of spin states. We find that the superpositions of Dicke states perform better than Dicke states themselves in phase esitmation. Although the spin coherent states themselves only have a shot-noise limit phase sensitivity, their superpositions may reach the Heisenberg limit.
The Fisher information FFF gives a limit to the ultimate precision achievable in a phase estimation protocol. It has been shown recently that the Fisher information for a linear two-mode interferometer cannot exceed the number of particles if the input state is separable. As a direct consequence, with such input states the shot-noise limit is the ultimate limit of precision. In this work, we go a step further by deducing bounds on FFF for several multiparticle entanglement classes. These bounds imply that genuine multiparticle entanglement is needed for reaching the highest sensitivities in quantum interferometry. We further compute similar bounds on the average Fisher information F‾\overline{F}F for collective spin operators, where the average is performed over all possible spin directions. We show that these criteria detect different sets of states and illustrate their strengths by considering several examples, also using experimental data. In particular, the criterion based on F‾\overline{F}F is able to detect certain bound entangled states.
Entanglement is the key quantum resource for improving measurement sensitivity beyond classical limits. However, the production of entanglement in mesoscopic atomic systems has been limited to squeezed states, described by Gaussian statistics. Here, we report on the creation and characterization of non-Gaussian many-body entangled states. We develop a general method to extract the Fisher information, which reveals that the quantum dynamics of a classically unstable system creates quantum states that are not spin squeezed but nevertheless entangled. The extracted Fisher information quantifies metrologically useful entanglement, which we confirm by Bayesian phase estimation with sub-shot-noise sensitivity. These methods are scalable to large particle numbers and applicable directly to other quantum systems.
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Quantum Fisher information, which plays a central role in parameter estimation theory, is a useful quantity for metrological applications. We use quantum Fisher information to distinguish and characterize behaviors of the evolved state for Bose-Einstein condensates in a symmetric double well, which display a classical bifurcation and a transition from Josephson oscillation to self-trapping. We find that the precision of the parameter estimation on the evolved state in the unstable regime is much higher than that in the stable regime, which lies at the shot-noise level. We also show that the parameter sensitivity of the evolved state approaches the Heisenberg limit in the Josephson oscillation regime, while it is just scaled as the shot-noise limit in the self-trapping regime. Finally, we discuss the relation between the parameter sensitivity of the evolved state and the entanglement entropy.