Cazul pur discret

In continuare vom considera cazul unei observabile exacte cu spectru pur discret ( $\Omega =\Bbb{Z}$) $\{\widehat{E}_i\}_{i\in \Bbb{Z}}$, pentru care vom aplica procedura standard de fuzzificare [240,14]:

$$\widehat{F}_\alpha =\sum\limits_{m\in \mathbb{Z}}{\omega _{\alpha m}} \widehat{E}_m\quad \forall \alpha \in \Bbb{Z}$$ (4.3.24)

$$\widehat{F}(B)=\sum\limits_{\alpha \in B}\sum\limits_{m\in \mathbb{Z}}{ \omega _{\alpha m}}\widehat{E}_m$$ (4.3.25)

unde constantele pozitive $\left\{ \omega _{\alpha m}\right\} _{\alpha ,m\in \Bbb{Z}}$ respecta relatia (matrice simplu stochastica):

$$\sum\limits_{\alpha \in \mathbb{Z}}\omega _{\alpha m}=1\quad \forall m\in \Bbb{Z}$$ (4.3.26)

si pot depinde de unii parametri de fuzzificare (cum ar fi $\sigma$ pe care, deocamdata nu ii mentionam). Pornind de la (4.2.14), luam $N_0=1$, iar pentru $\nu$ luam cazul masurii Heaviside, in care dependenta de $\omega$ dispare. Atunci obtinem rezultatul clasic din [14,161]:

$$\mathcal{E}^{*}(B,\hat{\rho})=\sum\limits_{m\in B}\widehat{\mathcal{A}}_m \hat{\rho}\widehat{\mathcal{A}}_m^{+}$$ (4.3.27)

unde:

$$\sum\limits_{m\in \mathbb{Z}}\widehat{\mathcal{A}}_m^{+}\widehat{\mathcal{A}} _m=\widehat{1}$$

Din (4.2.5), (4.3.24) si (4.3.27) se obtine:

$$\widehat{\mathcal{A}}_m\widehat{\mathcal{A}}_m^{+}=\sum\limits_{k\in \mathbb{Z}}{\omega _{mk}}\widehat{E}_k=\widehat{F}_m$$

unde consideram solutia ne-complexa:

$$\widehat{\mathcal{A}}_m=\widehat{\mathcal{A}}_m^{+}=\sum\limits_{k\in \mathbb{Z}}\sqrt{{\omega _{mk}}}\widehat{E}_k=\sqrt{\widehat{F}_m}$$ (4.3.28)

astfel ca (4.3.27) devine:

$$\mathcal{E}^{*}(B,\hat{\rho})=\sum\limits_{m\in B}\sum\limits_{k,... ...athbb{Z}}\sqrt{{\omega _{mk}\omega _{mn}}}\widehat{E}_k\hat{\rho}\widehat{E }_n$$ (4.3.29)

$$\mathcal{E}^{*}(B,\hat{\rho})=\sum\limits_{m\in B}\sqrt{\widehat{F}_m}\hat{ \rho}\sqrt{\widehat{F}_m}$$ (4.3.30)

Rezultatul (4.3.30) este de multe ori citat ca obiectivul principal al OQP (vezi pagina 138 in [53]). Desigur, in situatia exacta, cand ${\omega _{mk}=\delta _{mk}}$, se obtine rezultatul obisnuit al lui von Neumann [38]:

$$\mathcal{E}_{vN}^{*}(B,\hat{\rho})=\sum\limits_{m\in B}\widehat{E}_m\hat{\rho }\widehat{E}_m$$ (4.3.31)

Probabilitatile obtinute la masurarea unei observabile fuzzy $\{\widehat{F} _\alpha \}_{\alpha \in \Bbb{Z}}$ pentru un sistem in starea $\Bbb{{\hat{\rho} }}$ sunt:

$$p_\alpha =Tr(\widehat{F}_\alpha \Bbb{{\hat{\rho}})}=\sum\limits_{... ...{E}_m\hat{\rho})=\sum\limits_{m\in \mathbb{Z}}\omega _{\alpha m}\hat{\rho}_{mm}$$ (4.3.32)

iar aplicatia dinamica in imagine Schrodinger se poate scrie exprimand ( 4.3.32) in termenii unei aplicatii complet pozitive, similara instrumentului dual (4.3.27):

$$\widehat{\rho }\mapsto \widehat{\rho }_{post}^{(E)}=\mathcal{E}(\... ...Z}}\widehat{\mathcal{A}}_m^{(E)} \widehat{\rho }\widehat{\mathcal{A}}_m^{(E)+}$$

$$p_\alpha =Tr(\widehat{E}_\alpha \widehat{\rho }_{post}^{(E)})=Tr(... ...\widehat{\rho }\widehat{\mathcal{A}}_m^{(E)+})\quad \forall \alpha \in \Bbb{Z}$$

$$\sum\limits_{m\in \mathbb{Z}}\omega _{\alpha m}\rho _{mm}=\sum\li... ...ehat{\mathcal{A}}_m^{(E)+}\vert\alpha \rangle \quad \forall \alpha \in \Bbb{Z }$$ (4.3.33)

In (4.3.33) identificam:

$$\widehat{\mathcal{A}}_m^{(E)} = \sum\limits_{k\in \mathbb{Z}}\sqrt{{\omega _{km}}}\vert k\rangle \langle m\vert$$ (4.3.34)

$$\widehat{\mathcal{A}}_m^{(E)+} = \sum\limits_{k\in \mathbb{Z}}\sqrt{{\omega _{km}}}\vert m\rangle \langle k\vert$$

care sunt, desigur, diferite de (4.3.28).

Fie $\widehat{\mathcal{F}}_d$ operatorul pentru care:

$$\omega _{km}=\langle m\vert\widehat{\mathcal{F}}_d^{+}\vert k\rangle \langle k\vert \widehat{\mathcal{F}}_d\vert m\rangle$$ (4.3.35)

Luam si aici solutia ne-complexa pozitiva: $\langle k\vert\widehat{\mathcal{F}} _d\vert m\rangle =\sqrt{\omega _{km}}$, astfel ca putem scrie:

$$\widehat{\rho }_{post}^{(E)}=\widehat{\mathcal{F}}_d\sum\limits_{... ...mathbb{Z}}\widehat{E}_m\widehat{\rho }\widehat{E}_m\widehat{\mathcal{F}} _d^{+}$$ (4.3.36)

care poate fi adusa in forma (4.3.27) pentru:

$$\widehat{\mathcal{A}}_m^{(E)} = \widehat{\mathcal{F}}_d\widehat{E}_m$$ (4.3.37)

$$\widehat{\mathcal{A}}_m^{(E)+} = \widehat{E}_m\widehat{\mathcal{F}}_d^{+}$$

Cazul ideal de tip von Neumann se obtine si aici, pentru $\widehat{\mathcal{F }}_d=\widehat{1}$.