Variance is a ubiquitous quantity in quantum information theory. Given a basis, we consider the averaged variances of a fixed diagonal observable in a pure state under all possible permutations on the components of the pure state and call it symmetrized variance. Moreover, we work out the analytical expression of the symmetrized variance and find that such an expression is in the factorized form where two factors separately depend on the diagonal observable and quantum state. By shifting the factor corresponding to the diagonal observable, we introduce the notion referred to as standard symmetrized variance for the pure state which is independent of the diagonal observable. We then extend the standard symmetrized variance to mixed states in three different ways, which characterize the uncertainty, the coherence, and the coherence of assistance, respectively. These quantities are evaluated analytically and the relations among them are established. In addition, we show that standard symmetrized variance is also an entanglement measure for bipartite systems. In this way, these different quantumnesses of quantum states are unified by the variance.
Quantum coherence is a fundamental common trait of quantum phenomena, from the interference of matter waves to quantum degeneracy of identical particles. Despite its importance, estimating and measuring quantum coherence in generic, mixed many-body quantum states remains a formidable challenge, with fundamental implications in areas as broad as quantum condensed matter, quantum information, quantum metrology, and quantum biology. Here, we provide a quantitative definition of the variance of quantum coherent fluctuations (the quantum variance) of any observable on generic quantum states. The quantum variance generalizes the concept of thermal de Broglie wavelength (for the position of a free quantum particle) to the space of eigenvalues of any observable, quantifying the degree of coherent delocalization in that space. The quantum variance is generically measurable and computable as the difference between the static fluctuations and the static susceptibility of the observable; despite its simplicity, it is found to provide a tight lower bound to most widely accepted estimators of ``quantumness'' of observables (both as a feature as well as a resource), such as the Wigner-Yanase skew information and the quantum Fisher information. When considering bipartite fluctuations in an extended quantum system, the quantum variance expresses genuine quantum correlations among the two parts. In the case of many-body systems, it is found to obey an area law at finite temperature, extending therefore area laws of entanglement and quantum fluctuations of pure states to the mixed-state context. Hence the quantum variance paves the way to the measurement of macroscopic quantum coherence and quantum correlations in most complex quantum systems.
We investigate the ultimate precision limits for quantum phase estimation in terms of the coherence,
C, of the probe. For pure states, we give the minimum estimation variance attainable,
V(C), and the optimal state, in the asymptotic limit when the probe system size,
n, is large. We prove that pure states are optimal only if
C scales as
n with a sufficiently large proportionality factor, and that the rank of the optimal state increases with decreasing
C, eventually becoming full-rank. We show that the variance exhibits a Heisenberg-like scaling,
V(C)∼an/C2, where
an decreases to
π2/3 as
n increases, leading to a dimension-independent relation.