Fisher information and uncertainty complementarity

Z. Hradil, Jaroslav Řeháček

Shengjun Wu

J. S. Dehesa, Rodolfo O. Esquivel, Angel Ricardo Plastino +1 more

Shuangshuang Fu, Xiaohui Li

In this paper, employing the metric-adjusted skew information, we bring up a family of decompositions of the total uncertainty as quantified by the variance into the quantum uncertainty and classical uncertainty. Based on the decompositions of uncertainty, we further establish a family of complete complementarity relations for predictability, interference, and correlation in multipath interferometers. Explicitly, we show that the predictability can be quantified by the (reverse) total uncertainty, the interference can be quantified by the quantum uncertainty, while the correlation can be quantified by the classical uncertainty. The family of complementarity relations established in this paper generalize previous characterizations of interference in multipath interferometers via the quantum Fisher information [Y. Sun et al., Phys. Rev. A 108, 062416 (2023)]. As an illustration, we calculate the relevant quantities for some extremal quantum states, and also give explicitly the complementarity relations in two-path interferometers, which demonstrate the validity of the quantifiers. We compare our results with two related complementarity relations established recently and highlight the main differences. The results obtained in this work reveal the close connections between uncertainty and complementarity.

Yuan Sun, Nan Li, Shunlong Luo

Wave-particle duality, as complementarity displayed in interferometry, has been investigated extensively from both theoretical and experimental perspectives. In this work, we quantify coherence in multipath interference via quantum Fisher information and illuminate its basic properties. We prove that any rational quantifier of coherence satisfying monotonicity under incoherent operations is nonincreasing in the presence of path detectors. This quantitatively implies that increase on the path information will lead to the loss of coherence or interference (wave feature), which is consistent with wave-particle duality. Furthermore, we provide an operational illustration of wave feature as quantum Fisher information of phase shift parameters encoded in each interference path. By dividing the variance of a quantum state relative to a von Neumann measurement into quantum uncertainty and classical uncertainty, we establish a coherence-predictability-correlations triality relation, with coherence (wave feature) quantified by quantum Fisher information, predictability quantified by purity of a classical probability distribution, and correlations quantified by classical uncertainty relative to the von Neumann measurement determined by the interference paths. Finally, we make a comparison between this triality relation and the wave-particle-mixedness triality relation in the literature.

Jun Zhang, Wei Jia-ning, Zhoubo Duan +2 more

Abstract Wigner–Yanase skew information could quantify the quantum uncertainty of the observables that are not commuting with a conserved quantity. We present the uncertainty principle for two successive projective measurements in terms of Wigner–Yanase skew information based on a single quantum system. It could capture the incompatibility of the observables, i.e. the lower bound can be nontrivial for the observables that are incompatible with the state of the quantum system. Furthermore, the lower bound is also constrained by the quantum Fisher information. In addition, we find the complementarity relation between the uncertainties of the observable which operated on the quantum state and the other observable that performed on the post-measured quantum state and the uncertainties formed by the non-degenerate quantum observables performed on the quantum state, respectively.