Maximum Fisher information in mixed state quantum systems. (English) Zbl 1045.62122

Luati, Alessandra

Ann. Stat. 32, No. 4, 1770-1779 (2004).

Summary: We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists [C. W. Helstrom, Phys. Lett. A 25, 101–102 (1967)] who defined a quantum counterpart of the classical Fisher information, which has been found to constitute an upper bound for the classical information itself [S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439–3443 (1994; Zbl 0973.81509)]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A, Math. Gen. 33, No. 24, 4481–4490 (2000; Zbl 1004.81006)].
We show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive an expression of the maximum Fisher information achievable and its relation with that attainable in pure states.

MSC:

62P35	Applications of statistics to physics
81P05	General and philosophical questions in quantum theory
62B10	Statistical aspects of information-theoretic topics
62F99	Parametric inference

Keywords:

parametric quantum models; Fisher information; Helstrom information; symmetric logarithmic derivatives; pure states; mixed states

Citations:

Zbl 0973.81509; Zbl 1004.81006

References:

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