Abstract By comparing measurement-induced classical Fisher information of parameterized quantum states with quantum Fisher information,we study the notion of Fisher concord (as abbreviation of the concord between the classical and the quantum Fisher information), which is an information-theoretic measure of quantum states and quantum measurements based on both classical and quantum Fisher information. Fisher concord is defined by multiplying the inverse square root of quantum Fisher information matrix to measurement-induced classical Fisher information matrix on both sides, and quantifies the relative accessibility of parameter information from quantum measurements (alternatively, the efficiency of quantum measurements in extracting parameter information). It reduces to the ratio of the classical Fisher information to quantum Fisher information in any single parameter scenario. In general, Fisher concord is a symmetric matrix which depends on both quantum states and quantum measurements. Some basic properties of Fisher concord are elucidated. The significance of Fisher concord in quantifying the interplay between classicality and quantumness in parameter estimation and in characterizing the ef- ficiency of quantum measurements are illustrated through several examples, and some information conservation relations in terms of Fisher concord are exhibited.
Abstract By comparing measurement-induced classical Fisher information of parameterized quantum states with quantum Fisher information,we study the notion of Fisher concord (as abbreviation of the concord between the classical and the quantum Fisher information), which is an information-theoretic measure of quantum states and quantum measurements based on both classical and quantum Fisher information. Fisher concord is defined by multiplying the inverse square root of quantum Fisher information matrix to measurement-induced classical Fisher information matrix on both sides, and quantifies the relative accessibility of parameter information from quantum measurements (alternatively, the efficiency of quantum measurements in extracting parameter information). It reduces to the ratio of the classical Fisher information to quantum Fisher information in any single parameter scenario. In general, Fisher concord is a symmetric matrix which depends on both quantum states and quantum measurements. Some basic properties of Fisher concord are elucidated. The significance of Fisher concord in quantifying the interplay between classicality and quantumness in parameter estimation and in characterizing the ef- ficiency of quantum measurements are illustrated through several examples, and some information conservation relations in terms of Fisher concord are exhibited.
In classical statistics, the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities. In quantum theory, this uniqueness breaks down, and there are many natural quantum analogues of the Fisher information, among which two particular versions distinguish themselves by their intuitive and informational significance: The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement, and is defined via the square root of the density operator. The second arises from Helstrom's study of quantum detection in 1967, and is defined via the symmetric logarithmic derivative. The aim of this paper is to compare these two versions of quantum Fisher information, and to establish two informational inequalities relating them.
Informationally complete measurements form the foundation of universal quantum state reconstruction, while quantum parameter estimation is based on the local structure of the manifold of quantum states. Here we establish a general link between these two aspects, in the context of a single informationally complete measurement, by employing a suitably adapted operator frame theory. In particular, we bound the ratio between the classical and quantum Fisher information in terms of the spectral decomposition of the associated frame operator, and connect these bounds to the optimal and least optimal directions for parameter encoding. The geometric and operational characterization of information extraction thus obtained reveals the fundamental tradeoff imposed by informational completeness on local quantum parameter estimation.
In classical statistics,the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities.In quantum theory,this uniqueness breaks down,and there are many natural quantum analogues of the Fisher information,among which two particular versions distinguish themselves by their intuitive and informational significance:The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement,and is defined via the square root of the density operator. The second arises from Helstrom's study of quantum detection in 1967,and is defined via the symmetric logarithmic derivative.The aim of this paper is to compare these two versions of quantum Fisher information,and to establish two informational inequalities relating them.
Quantum Fisher information, as an intrinsic quantity for quantum states, is a central concept in quantum detection and estimation. When quantum measurements are performed on quantum states, classical probability distributions arise, which in turn lead to classical Fisher information. In this paper, we exploit the classical Fisher information induced by quantum measurements and reveal a rich hierarchical structure of such measurement-induced Fisher information. We establish a general framework for the distribution and transfer of the Fisher information. In particular, we illustrate three extremal distribution types of the Fisher information: the locally owned type, the locally inaccessible type, and the fully shared type. Furthermore, we indicate the significant role played by the distribution and flow of the Fisher information in some physical problems, e.g., the non-Markovianity of open quantum processes, the environment-assisted metrology, the cloning and broadcasting, etc.
Two classically equivalent expressions of mutual information of probability distributions (classical bipartite states) diverge when extended to quantum systems, and this difference has been employed to define quantum discord, a quantifier of quantum correlations beyond entanglement. Similarly, equivalent expressions of classical Fisher information of parameterized probability distributions diverge when extended to quantum states, and this difference may be exploited to characterize the complex nature of quantum states. By complexity of quantum states, we mean some hybrid nature which intermingles the classical and quantum features. It is desirable to quantify complexity of quantum states from various perspectives. In this work, we pursue the idea of discord and introduce an information-theoretic quantifier of complexity for quantum states (relative to the Hamiltonian that drives the evolution of quantum systems) via the notion of Fisher discord, which is defined by the difference between two important versions of quantum Fisher information: the quantum Fisher information defined via the symmetric logarithmic derivatives and the Wigner-Yanase skew information defined via the square roots of quantum states. We reveal basic properties of the quantifier of complexity, and compare it with some other quantifiers of complexity. In particular, we show that equilibrium states (or stable states, which commute with the Hamiltonian of the quantum system) and all pure states exhibit zero complexity in this setting. As illustrations, we evaluate the complexity for various prototypical states in both discrete and continuous-variable quantum systems.
With the development of quantum-information theory, there has been a flurry of investigations of quantum non-Markovian dynamics, and several significant measures for such dynamics have been proposed from various perspectives, such as the breakdown of dynamical divisibility, increase in the distinguishability between quantum states, increase in correlations between the system and an arbitrary ancillary, and so on. Motivated by the idea of exploiting the information content of parameters encoded in initial states, we propose a conceptually simple and physically intuitive characterization for non-Markovianity with the help of a quantum-Fisher-information matrix. The basic features are illustrated through several examples, and relations with other approaches are elucidated. A hierarchial aspect of quantum non-Markovianity is revealed.