Post-processing minimal joint observables

A finite set of quantum observables (positive operator valued measures) is called compatible if
these observables are marginals of a some observable, called a joint observable of them. For a given
set of compatible observables, their joint observable is in general not unique and it is desirable
to take a minimal joint observable in the post-processing order since a less informative observable
disturbs the system less. We address the question of the minimality of finite-outcome joint
observables and prove that any joint observable is lower bounded by a minimal joint observable in
the post-processing order. We also give characterizations of the minimality of a joint observable
that can be checked by finite-step algorithms and apply them to the case of non-commuting dichotomic
qubit observables.

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