of the fluctuations) and its von Neumann entropy $S(\overline \rho) = -\tr(\overline \rho \ln \overline \rho)$. The results are compared with the coherence and the von Neumann entropy of the reduced density matrix $\rho_0$ for $X_0$ without fluctuation, in order to distinguish the effects due to the quantum entanglement (present in the two cases) from the effects of the chaos (present only with the fluctuations). Some results are drawn figure \[cohe\].

![\[cohe\] Coherence and von Neumann entropy of the spin mixed state of the fermionic string, for $X_0$ without fluctuation ($\rho_0$) and for $X$ with fluctuations ($\overline \rho$), with fuzzy sphere, torus and hyperboloid. The initial fluctuations are choosen following a gaussian law with $\sigma=0.01$. The number of D0-branes is $N=11$ (for the fuzzy hyperboloid the cutoff is $N=11$). The vertical line indicates the horizon of coherence. The initial condition for the string is $|\psi(t=0) {\rangle \hspace{-0.2em} \rangle}= \frac{1}{\sqrt 2}(|\uparrow \rangle+|\downarrow \rangle) \otimes (1,0,...,0)$.](coheSphere.eps "fig:"){width="6.4cm"} ![\[cohe\] Coherence and von Neumann entropy of the spin mixed state of the fermionic string, for $X_0$ without fluctuation ($\rho_0$) and for $X$ with fluctuations ($\overline \rho$), with fuzzy sphere, torus and hyperboloid. The initial fluctuations are choosen following a gaussian law with $\sigma=0.01$. The 