Theoretical and Mathematical Physics

, Volume 143, Issue 2, pp 681–688

Quantum versus classical uncertainty

  • S. L. Luo

DOI: 10.1007/s11232-005-0098-6

Cite this article as:
Luo, S.L. Theor Math Phys (2005) 143: 681. doi:10.1007/s11232-005-0098-6


The uncertainty of an observable in a quantum state is usually described by variance. This description is well suited when the states are pure. But when the states are mixed, things become subtle, and the variance is a hybrid of quantum and classical uncertainties. Motivated by the notion of Fisher information in statistical inference, we establish a decomposition of the variance into quantum and classical parts. The key observation is that the Wigner-Yanase skew information (a distinguished version of quantum Fisher information) can be interpreted as a measure of quantum uncertainty. We also establish a decomposition of the conventional covariance into quantum and classical parts. The results provide a new perspective for understanding uncertainty and correlation and are used to quantify entanglement, as well as to establish a new uncertainty relation in purely quantum terms.


uncertaintyclassical uncertaintyquantum uncertaintyskew informationquantum correlationentanglementuncertainty principle

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. L. Luo
    • 1
    • 2
  1. 1.Academy of Mathematics and Systems Sciences, Chinese Academy of SciencesBeijingP. R. China
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada