We address nearly pure quantum statistical models, i.e. situations where the information about a parameter is encoded in pure states weakly perturbed by the mixing with a parameter independent state, mimicking a weak source of noise. We show that the symmetric logarithmic derivative is left unchanged, and find an approximate analytic expression for the quantum Fisher information (QFI) which provides bounds on how much a weak source of noise may degrade the QFI.

International Journal of Quantum InformationJan 28, 20200 citations

International Journal of Theoretical Physics

Quantum Fisher Information in the Symmetric Pure Three-Qubit State

Desheng Liu, Juan Du, Guoqiang Huang

Published Feb 12, 2012• 4 citations

Publisher

DOI: 10.1007/s10773-012-1090-1

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About the quantum Fisher information of nearly pure quantum statistical models

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\xi _{2}^{2}

. 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.

We study the reciprocal of the mean quantum Fisher information (RMQFI),

χ^2

for general three qubit states, having graph and hypergraph states as special cases, for identifying genuine multi party entanglement characterized by

χ^2 <1

. We demonstrate that the most symmetric graph state and the GHZ state have the lowest RMQFI values leading to the highest statistical speed showing that both these states attain the Heisenberg limit in phase sensitivity. Unlike the GHZ state, graph states have the same RMQFI values for measurement through different parameters, a property shared by the hypergraph states. Three qubit graph and hypergraph states can violate Bell's inequality as

F_Q > N

. Both the GHZ state and the most symmetric graph state have the highest concurrence equalling 3 and the maximum QFI values.