Optimal quantum estimation in spin systems at criticality

It is a general fact that the coupling constant of an interacting many-body Hamiltonian does not correspond to any observable and one has to infer its value by an indirect measurement. For this purpose, quantum systems at criticality can be considered as a resource to improve the ultimate quantum limits to precision of the estimation procedure. In this paper, we consider the one-dimensional quantum Ising model as a paradigmatic example of a many-body system exhibiting criticality, and derive the optimal quantum estimator of the coupling constant varying size and temperature. We find the optimal external field, which maximizes the quantum Fisher information of the coupling constant, both for few spins and in the thermodynamic limit, and show that at the critical point a precision improvement of order $L$ is achieved. We also show that the measurement of the total magnetization provides optimal estimation for couplings larger than a threshold value, which itself decreases with temperature.

[1]  Shun-ichi Amari,et al.  Methods of information geometry , 2000 .

[2]  W. Wootters.  Statistical distance and Hilbert space , 1981 .

[3]  Paolo Zanardi,et al.  Information-theoretic differential geometry of quantum phase transitions. , 2007, Physical review letters.

[4]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[5]  M. Nielsen,et al.  Entanglement in a simple quantum phase transition , 2002, quant-ph/0202162.

[6]  J. P. Barjaktarevic,et al.  Fidelity and quantum phase transitions , 2007, cond-mat/0701608.

[7]  Marco G. Genoni,et al.  Optimal estimation of entanglement , 2008, 0804.1705.

[8]  H. Sommers,et al.  Bures volume of the set of mixed quantum states , 2003, quant-ph/0304041.

[9]  M. Cozzini,et al.  Quantum fidelity and quantum phase transitions in matrix product states , 2007 .

[10]  L. L. Cam,et al.  Asymptotic Methods In Statistical Decision Theory , 1986 .

[11]  P. Zanardi,et al.  Quantum phase transitions and quantum fidelity in free fermion graphs , 2006, quant-ph/0608059.

[12]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[13]  G. Milburn,et al.  Generalized uncertainty relations: Theory, examples, and Lorentz invariance , 1995, quant-ph/9507004.

[14]  G. Vidal,et al.  Entanglement in quantum critical phenomena. , 2002, Physical review letters.

[15]  A. Osterloh,et al.  Scaling of entanglement close to a quantum phase transition , 2002, Nature.

[16]  Paolo Zanardi,et al.  Bures metric over thermal state manifolds and quantum criticality , 2007, 0707.2772.

[17]  Dorje C. Brody,et al.  Statistical geometry in quantum mechanics , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  Dorje C. Brody,et al.  Geometrization of statistical mechanics , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  N. Paunkovic,et al.  Ground state overlap and quantum phase transitions. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  A. Uhlmann.  The "transition probability" in the state space of a ∗-algebra , 1976 .

[21]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[22]  Paolo Zanardi,et al.  Quantum critical scaling of the geometric tensors. , 2007, Physical review letters.