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Quantum Information and Computation

Quantum fisher information for superpositions of spin states

XiongHeng-Na, MaJian, LiuWan-Fang, WANGXiao-Guang

Published May 1, 2010• 0 citations
Publisher

Abstract

In terms of quantum Fisher information, a quantity X2 was introduced by Pezze andSmerzi [Phys. Rev. Lett. 102, 100401 (2009)], which is a multiparticle entanglementmeasure, and provides a necessary...
DOI
10.5555/2011362.2011370

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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.
In terms of quantum Fisher information (QFI), a quantity χ 2 was introduced by Pezzé and Smerzi (Phys. Rev. Lett. 102 100401, 2009). They pointed out that the inequality χ 2<1 was a sufficient condition for multiparticle entanglement. For the two-qubit symmetric states, we found that the inequality χ 2<1 is a necessary and sufficient condition for entanglement and spin squeezing, and that χ 2 is equal to the second kind of spin squeezing parameter ξ22\xi _{2}^{2}ξ22​ . For the two-qubit asymmetric states, it is only a sufficient condition. In order to make it a necessary and sufficient condition, we extend the concept of the QFI and χ 2, and generalize the relations among the entanglement measurement, the spin squeezing parameters and χ 2 in symmetric pure states to arbitrary pure states.
Fisher information, which lies at the heart of parameter estimation theory, was recently found to have a close relation with multipartite entanglement [L. Pezz\'e and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009)]. We use Fisher information to distinguish and characterize behaviors of ground state of the Lipkin-Meskhov-Glick model, which displays a second-order quantum phase transition between the broken and symmetric phases. We find that the parameter sensitivity of the system attains the Heisenberg limit in the broken phase, while it is just around the shot-noise limit in the symmetric phase. Based on parameter estimation, Fisher information provides us a useful approach to the quantum phase transition.