  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 (of discord type) 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.
