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Non-Commutative Extension of Information Geometry II

In: Quantum Communication, Computing, and Measurement

Author

Listed:
  • Hiroshi Hasegawa

    (Tamagawa University Tamagawa-gakuen, Research Center for Quantum Communications)

  • Dénes Petz

    (Technical University of Budapest, Department of Mathematical Analysis)

Abstract

The Fisher information provides a canonical Riemannian metric in the geometric approach to classical statistics. It seems that the quantum analogue of the Fisher information is not uniquely defined, and it is necessary to study the possible candidates and to compare them on physical grounds. Description of monotone metrics under coarse graining has been given by Petz and this class of metrics fixes many candidates. Here we show that the skew information $${I_p}\left( {\rho ,K} \right) \equiv - \frac{1}{2}Tr\left[ {{\rho ^p},K} \right]\left[ {{\rho ^{1 - p}},K} \right]$$ first introduced by Wigner, Yanase and Dyson (WYD) many years ago yields a monotone metric for all values of p; -1 ≤ p ≤ 2(for p = 0,1 under a proper limiting procedure and beyond the limits with a change of the sign of I p ). Furthermore, we argue that the symmetry between I p and I1-p is identical to the quantum version of Amari’s duality concept for smooth statistical manifolds.

Suggested Citation

  • Hiroshi Hasegawa & Dénes Petz, 1997. "Non-Commutative Extension of Information Geometry II," Springer Books, in: O. Hirota & A. S. Holevo & C. M. Caves (ed.), Quantum Communication, Computing, and Measurement, pages 109-118, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-5923-8_12
    DOI: 10.1007/978-1-4615-5923-8_12
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