Direct approach to quantum extensions of Fisher information

Anonymous, Jukka Kiukas, Daniel Klaus Burgarth +1 more

Liang Cai

Motivated by the direct approach to quantum extensions of Fisher information in Chen and Luo [Front. Math. China 2 (2007) 359], we introduce a further generalized version of Wigner–Yanase–Dyson information, and investigate its fundamental properties including positivity, convexity and super-additivity, etc. As an application, we present a generalized uncertainty relation.

Marcin Makowski, Edward W. Piotrowski, Piotr Frąckiewicz +1 more

The principle of minimum Fisher information states that in the set of acceptable probability distributions characterizing the given system, it is best done by the one that minimizes the corresponding Fisher information. This principle can be applied to transaction processes, the dynamics of which can be interpreted as the market tendency to minimize the information revealed about itself. More information involves higher costs (information is physical). The starting point for our considerations is a description of the market derived from the assumption of minimum Fisher information for a strategy with a fixed financial risk. Strategies of this type that minimize Fisher information overlap with the well-known eigenstates of a the quantum harmonic oscillator. The analytical extension of this field of strategy to the complex vector space (traditional for quantum mechanics) suggests the study of the interference of the oscillator eigenstates in terms of their minimization of Fisher information. It is revealed that the minimum value of Fisher information of the superposition of the two strategies being the ground state and the second excited state of the oscillator, has Fisher information less than the ground state of the oscillator. Similarly, less information is obtained for the system of strategies (the oscillator eigenstates) randomized by the Gibbs distribution. We distinguish two different views on the description of Fisher information. One of them, the classical, is based on the value of Fisher information. The second, we call it transactional, expresses Fisher information from the perspective of the constant risk of market strategies. The orders of the market strategies derived from these two descriptions are different. From a market standpoint, minimizing Fisher information is equivalent to minimizing risk.

Shunlong Luo

Xin-Zhu Liu, Jun-Li Jiang, Yan-Han Yang +4 more

Fisher information plays a central role in statistics and quantum metrology, providing the basis for the celebrated Cramér-Rao bound. In this work, we introduce a new information measure based on higher-order Fisher information and show that it naturally leads to a generalized uncertainty relation for parameter estimation, which can be regarded as an extension of the Cramér-Rao bound. As an application, we analyze the case of quantum phase estimation with a single qubit and compare our theoretical bounds with the well-known established hierarchical bounds. Finally, we experimentally validate the proposed framework using a photonic platform.

Hiroshi Nagaoka