  Which quantum states minimise the unavoidable uncertainty arising from the
non-commutativity of two observables? The immediate answer to such a question
is: it depends. Due to the plethora of uncertainty measures there are many
answers. Here, instead of restricting our study to a particular measure, we
present plausible axioms for the set $\mathcal{F}$ of bona-fide
information-theoretic uncertainty functions. Then, we discuss the existence of
states minimising uncertainty with respect to all members of $\mathcal{F}$,
i.e., universal minimum uncertainty states (MUS). We prove that such states do
not exist within the full state space and study the effect of classical noise
on the structure of minimum uncertainty states. We present an explicit example
of a qubit universal MUS that arises when purity is constrained by introducing
a threshold amount of noise. For higher dimensional systems we derive several
no-go results limiting the existence of noisy universal MUS. However, we
conjecture that universality may emerge in an approximate sense. We conclude by
discussing connections with thermodynamics, and highlight the privileged role
that non-equilibrium free energy $F_2$ plays close to equilibrium.
