Exemplu

Pentru (4.3.40) avem: $M_1^{(E)}=M_1$, dar pentru $n>1$, $M_n^{(E)}\neq M_n$. Entropiile (4.4.48) si (4.4.49) devin:

$$S^{(O)}=1-\sum\limits_{k\in \mathbb{Z}}\sum\limits_{m\in \mathbb{Z}}[\rho _{k,k+2m}\cdot \rho _{k+2m,k}\cdot \exp (-2\frac{m^2}{\sigma ^2})+$$

$$+(\frac{\Psi _{1/2}(\sigma )}{\Psi _0(\sigma )})^2\rho _{k,k+2m+1}\cdot \rho _{k+2m+1,k}\cdot \exp (-2\frac{(m+\frac 12)^2}{\sigma ^2})]$$

$$S^{(E)}=1-\sum\limits_{k\in \mathbb{Z}}\sum\limits_{m\in \mathbb{Z}}[\rho _{k,k}\cdot \rho _{k+2m,k+2m}\cdot \exp (-2\frac{m^2}{\sigma ^2})+$$

$$+(\frac{\Psi _{1/2}(\sigma )}{\Psi _0(\sigma )})^2\rho _{k,k}\cdot \rho _{k+2m+1,k+2m+1}\cdot \exp (-2\frac{(m+\frac 12)^2}{\sigma ^2})]$$

Pentru o stare initiala de tip fuzzy:

$$\widehat{\rho }=\widehat{\mathcal{F}}_d^{(\alpha )}\vert a\rangle... ...ehat{\mathcal{F}}_d^{(\alpha )+}\quad a\in \mathbb{Z},\alpha \in \mathbb{R}_{+}$$ (4.4.50)

(4.4.48) si (4.4.49) sunt egale:

$$S^{(O)}=S^{(E)}=1-\sum\limits_{n,k\in \mathbb{Z}}(\sum\limits_{m\... ...ga _{mn}^{(\sigma )}}})^2{ \omega _{n,a}^{(\alpha )}\omega _{k,a}^{(\alpha )}}$$

care pentru (4.3.40) este:

$$1-\frac 1{\left( \Psi _0(\alpha )\right) ^2}[\Psi _0(\frac \alpha 2)\Psi _0(\frac{\alpha \cdot \sigma }{\sqrt{2\cdot (\alpha ^2+2\sigma ^2)}})+$$

$$+(\frac{\Psi _{1/2}(\sigma )}{\Psi _0(\sigma )})^2\Psi _{1/2}(\fr... ...Psi _{1/2}(\frac{\alpha \cdot \sigma }{\sqrt{2\cdot (\alpha ^2+2\sigma ^2) }})]$$

Utilizand aproximatiile de la (4.3.41), pentru o pozitie initiala exacta $(\alpha <0.2)$, rezultatul este independent de parametrul de fuzzificare $\sigma$, $S=0$ (starea ramane aproape pura), in timp ce pentru $\alpha >1.6$ (pozitie initiala fuzzy), rezultatul este $1-\frac \alpha { \sqrt{\pi }}$ pentru masuratori exacte $(\sigma <0.2)$ si $1-\frac \sigma { \sqrt{2\cdot (\alpha ^2+2\sigma ^2)}}$ pentru masuratori non-exacte (daca $\alpha \ll \sigma$ aceasta atinge valoarea maxima pentru (4.4.47), $S=\frac 12$)