  The voter model with multiple states has found applications in areas as
diverse as population genetics, opinion formation, species competition and
language dynamics, among others. In a single step of the dynamics, an
individual chosen at random copies the state of a random neighbor in the
population. In this basic formulation it is assumed that the copying is
perfect, and thus an exact copy of an individual is generated at each time
step. Here we introduce and study a variant of the multi-state voter model in
mean-field that incorporates a degree of imperfection or error in the copying
process, which leaves the states of the two interacting individuals similar but
not exactly equal. This dynamics can also be interpreted as a perfect copying
with the addition of noise; a minimalistic model for flocking. We found that
the ordering properties of this multi-state noisy voter model, measured by a
parameter $\psi$ in [0, 1], depend on the amplitude $\eta$ of the copying error
or noise and the population size N. In the case of perfect copying $\eta=0$ the
system reaches an absorbing configuration with complete order ($\psi=1$) for
all values of N. However, for any degree of imperfection $\eta>0$, we show that
the average value of $\psi$ at the stationary state decreases with N as
$\langle \psi \rangle \simeq 6/(\pi^2 \eta^2 N)$ for $\eta \ll 1$ and $\eta^2 N
\gtrsim 1$, and thus the system becomes totally disordered in the thermodynamic
limit $N \to \infty$. We also show that $\langle \psi \rangle \simeq 1-1.64 \,
\eta^2 N$ in the vanishing small error limit $\eta \to 0$, which implies that
complete order is never achieved for $\eta > 0$. These results are supported by
Monte Carlo simulations of the model, which allow to study other scenarios as
well.
