  A striking feature of standard quantum mechanics is its analogy with
classical fluid dynamics. In particular it is well known the Schr\"{o}dinger
equation can be viewed as describing a classical compressible and non-viscous
fluid, described by two (quantum) fluid fields ${\rho ,% \mathbf{V}} $, to be
identified with the quantum probability density and velocity field. This
feature has suggested the construction of a phase-space hidden-variable
description based on a suitable inverse kinetic theory (IKT; Tessarotto et al.,
2007). The discovery of this approach has potentially important consequences
since it permits to identify the classical dynamical system which advances in
time the quantum fluid fields. This type of approach, however requires the
identification of additional fluid fields. These can be generally identified
with suitable directional fluid temperatures $T_{QM,i}$ (for $i=1,2,3$), to be
related to the expectation values of momentum fluctuations appearing in the
Heisenberg inequalities. Nevertheless the definition given previously for them
(Tessarotto et al., 2007) is non-unique. In this paper we intend to propose a
criterion, based on the validity of a constant H-theorem, which provides an
unique definition for the quantum temperatures.
