Since the advent of quantum theory founded nearly a century ago, non-commutativity of quantum observables has undoubtedly been one of the major sources of troubles we face when we try to interpret their measurement outcomes in a sensible manner. This has naturally led to various attempts of "quantisation" of classical systems in terms of non-commuting Hilbert space operators, or conversely of "quasi-classical" interpretation of quantum systems in terms of commuting quantities familiar to us in classical theory.
For commuting quantum observables, "trivial" methods of quantisation and classicalisation are available, where the former is known as the functional calculus while the latter is known as the Born rule. These maps are both known to be characterised by (joint-)spectral measures ((J)SMs), and they are understood to be adjoint operations to each other. In this talk, we propose a novel approach to the method of quantisation and quasi-classicalisation meaningful even for non-commuting observables, by introducing the concept of "quasi-joint-spectral distributions (QJSDs)", which are intended as a non-commuting generalisation to the standard JSMs of commuting observables. Specifically, we see that there are inherent indefiniteness in the possible definition of QJSDs, and this leads to various candidates of quantisation/quasi-classicalisation, while still preserving their adjointness relations.
As applications, we consider quantum analogues of correlations and conditionings induced by QJSDs. Specifically, we point out that Aharonov's weak value can be understood as one realisation of quantum conditional expectations corresponding to a specific choice of the QJSDs. We moreover consider quantum analogues of the covariance inequality, and point out a trade-off relation between approximation of an observable and estimation of a parameter (uncertainty relation for approximation and estimation), where quantum conditional expectations (incl. the weak value) play an essential role.
 J. Lee and I. Tsutsui. "Uncertainty relations for approximation and estimation". Phys. Lett. A, 380:2045, 2016.
 J. Lee and I. Tsutsui. "Quasi-probabilities in Conditioned Quantum Measurement and a Geometric/Statistical Interpretation of Aharonov's Weak Value", PTEP (In press); arXiv:1607.06406.