Entanglement, weak values, and the precise inference of joint measurement outcomes for non-commuting observable pairs

The problem of inferring the outcome of a simultaneous measurement of two non-commuting observables is addressed. We show that for certain pairs with dense spectra, precise inferences of the measurement outcomes are possible in pre- and postselected ensembles, and if the selections involve entangled states with some other system. We show that the problem is related to the problem of assigning weak values to a continuous family of operators, and give explicit examples where this problem is solvable. A quantum-optical experiment is suggested and some foundational implications are briefly discussed.