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Fisher Information And Fidelity

Posted on:2010-12-17Degree:MasterType:Thesis
Country:ChinaCandidate:L XuFull Text:PDF
GTID:2190330332993246Subject:Theoretical Physics
Abstract/Summary:PDF Full Text Request
In recent years, a great efforts have been focused on the concept of fidelity in quantum information theory, which have a intrinsic relation with Fisher In-formation(FI) in estimation theory. And many works have applied the fidelity and fidelity susceptibility approach to analyze quantum phase transition. Thus in this thesis, we apply the approaches of fidelity and fidelity susceptibility to discuss quantum phase transitions(QPT) in Lipkin-Meshkov-Glick model and its scaling behaviors around the critical point.In Chapter 1, we mainly give some general backgrounds about FI in clas-sical statistical theory and some key concepts in quantum information theory, i.e., fidelity, fidelity susceptibility, etc. And retail their mechanism in analyzing Quantum phase transition.In Chapter 2, we review FI from the perspective of statistical distance. And we get its quantum analogues, Quantum Fisher Information(QFI) by symmet-ric logarithmic derivative(SLD) method and a heuristic method from the classic FI. And as a concept of statistical distance, we find its relation with the well known Bures distance, Hellinger distance. It is shown that, in classical limit, 4ds2CH=4dsB2=IF, and in quantum case, we just find that I=4dsB2 and J=4dsQH2.In Chapter 3, We introduce the concept of reduced fidelity and fidelity sus-ceptibility. We give a general formula for the RFS when the density matrix is block diagonal in 2x2 matrices. Then we introduce the Lipkin-Meshcov-Glick Model. In the thermodynamic limit, the RFS is divergent as (1-h)-in the bro ken phase (0<h≤1), where h is the effective transverse magnetic field. How-ever, it becomes zero in the entire symmetry phase (h>1). In the finite-size situation, it is not suitable to use the RFS for the isotropic case in the symmetry phase due to energy degeneracy. For the anisotropic case, by using the continu-ous unitary transformation(CUT), we find that the maximum ofχvs h diverges as N2/3 for an N-spin system.In Chapter 4, We generalize the results in chapter 3 by studying reduced den-sity matrix concerning arbitrary local system of LMG model instead the density matrix in 2x2 matrices in terms of fidelity susceptibility, which can be regarded as reduced fidelity susceptibility (RFS). We find the pseudocritical point of RFS shifts toward the critical point as N in the thermodynamic limit. And We also obtain the some size scaling behavior in terms of RFS with different size of local system.
Keywords/Search Tags:FI, QFI, Bures/Hellinger distance, fidelity, reduced fidelity, susceptibility, RFS, QPT, SLD, Scaling behavior
PDF Full Text Request
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