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 parametrized probability distributions diverge when extended to quantum states, and this difference may be exploited to characterize certain complex natures of quantum states. Here, 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 metrological 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 metrological complexity, and compare it with some other quantifiers of complexity. In particular, we show that stationary states (or equilibrium states of the quantum system) and all pure states exhibit zero metrological complexity in this setting. As illustrations, we evaluate the metrological complexity for various prototypical states in both discrete and continuous-variable quantum systems, and demonstrate its applications in detecting quantum phase transitions in the $XY$ spin model.

Close

PrevNext

View figure in article

• Quantum Fisher information
• Quantum foundations
• Quantum information theory