  Quantum measurements are our eyes to the quantum systems consisting of a
multitude of microscopic degrees of freedom. However, the intrinsic uncertainty
of quantum measurements and the exponentially large Hilbert space pose natural
barriers to simple interpretations of the measurement outcomes. We propose a
nonlinear "measurement energy" based upon the measurement outcomes and an
iterative effective-Hamiltonian approach to extract the most probable states
(maximum likelihood estimates) in an efficient and general fashion, thus
reconciling the non-commuting observables and getting more out of the quantum
measurements. We showcase the versatility and accuracy of our perspective on
random long-range fermion models and Kitaev quantum spin liquid models, where
smoking-gun signatures were lacking. Our study also paves the way towards
concepts such as nonlinear-operator Hamiltonian and applications such as parent
Hamiltonian reconstruction.
