  The von Neumann entanglement entropy is used to estimate the critical point
$h_c/J \simeq 0.143(3)$ of the mixed ferro-antiferromagnetic three-state
quantum Potts model $H = \sum_i [ J ( X_i X_{i+1}^{\,2} + X_i^{\,2} X_{i+1} ) -
h\, R_i ]$, where $X_i$ and $R_i$ are standard three-state Potts spin operators
and $J>0$ is the antiferromagnetic coupling parameter. This critical point
value gives improved estimates for two Kosterlitz-Thouless transition points in
the antiferromagnetic ($\beta < 0$) region of the $\Delta$--$\beta$ phase
diagram of the three-state quantum chiral clock model, where $\Delta$ and
$\beta$ are, respectively, the chirality and coupling parameters in the clock
model. These are the transition points $\beta_c \simeq - 0.143(3)$ at $\Delta =
\frac12$ between incommensurate and commensurate phases and $\beta_c \simeq -
7.0(1)$ at $\Delta = 0$ between disordered and incommensurate phases. The von
Neumann entropy is also used to calculate the central charge $c$ of the
underlying conformal field theory in the massless phase $h \le h_c$. The
estimate $c \simeq 1$ in this phase is consistent with the known exact value at
the particular point $h/J = -1$ corresponding to the purely antiferromagnetic
three-state quantum Potts model. The algebraic decay of the Potts spin-spin
correlation in the massless phase is used to estimate the continuously varying
critical exponent $\eta$.
