Annals of Statistics

Maximum Fisher information in mixed state quantum systems

Alessandra Luati

Full-text: Open access

Abstract

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, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. 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 [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490].

In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.

Article information

Source
Ann. Statist., Volume 32, Number 4 (2004), 1770-1779.

Dates
First available in Project Euclid: 4 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1091626187

Digital Object Identifier
doi:10.1214/009053604000000436

Mathematical Reviews number (MathSciNet)
MR2089142

Zentralblatt MATH identifier
1045.62122

Subjects
Primary: 62B05: Sufficient statistics and fields
Secondary: 62F10: Point estimation

Keywords
Parametric quantum models Fisher information Helstrom information symmetric logarithmic derivatives pure states mixed states

Citation

Luati, Alessandra. Maximum Fisher information in mixed state quantum systems. Ann. Statist. 32 (2004), no. 4, 1770--1779. doi:10.1214/009053604000000436. https://projecteuclid.org/euclid.aos/1091626187


Export citation

References

  • Amari, S.-I. and Nagaoka, H. (2000). Methods of Information Geometry. Oxford Univ. Press.
  • Barndorff-Nielsen, O. E. and Gill, R. D. (2000). Fisher information in quantum statistics. J. Phys. A 33 4481–4490.
  • Braunstein, S. L. and Caves, C. M. (1994). Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72 3439–3443.
  • Fujiwara, A. and Nagaoka, H. (1995). Quantum Fisher metric and estimation for pure state models. Phys. Lett. A 201 119–124.
  • Helstrom, C. W. (1967). Minimum mean-squared error of estimates in quantum statistics. Phys. Lett. A 25 101–102.
  • Helstrom, C. W. (1976). Quantum Detection and Estimation Theory. Academic Press, New York.
  • Holevo, A. S. (1982). Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam.
  • Luati, A. (2003). A real formula for transition probability.
  • Statistica To appear.