Entropii Shannon-von Neumann

Fie 2 sisteme cuantice care respecta conditiile din teorema Schmidt. Aplicand interpretarea Born pentru (3.2.6) rezulta ca probabilitatea ca sistemul compus sa se fie gasit in starea $\vert u_s\rangle \vert v_s\rangle$ in urma unei masuratori corespunzatoare este $\lambda _s.$ Daca atribuim aceleasi probabilitati si pentru starile sistemelor individuale, inseamna ca putem aplica teoria clasica a lui Shannon si in acest caz. Fie observabilele din $\mathcal{H}^{\prime }$ si $\mathcal{H}^{\prime \prime }$

$$\hat{A}=\sum_{n=1}^Na_n\hat{P}_n$$

$$\hat{B}=\sum_{p=1}^Mb_p\hat{Q}_p$$

Probabilitea de a obtine simultan valorile $a_s$, respectiv $b_p$ este

$$p(a_n,b_p)=\mathrm{tr}(\hat{P}_n\otimes \hat{Q}_p\hat{\rho})=$$

$$=\sum_{r,s=1}^N\sqrt{\lambda _s\lambda _r}\mathrm{tr(}\hat{P}_n\o... ..._p\vert u_s\rangle \langle u_r\vert\otimes \vert v_s\rangle \langle v_r\vert)=$$

$$=\sum_{s=1}^N\lambda _s\left\vert \alpha _{sn}\right\vert ^2\left\vert \beta _{sp}\right\vert ^2$$

unde

$$\left\vert \alpha _{sn}\right\vert ^2=\langle u_s\vert\hat{P}_n\vert u_s\rangle$$   si $$\quad \left\vert \beta _{sp}\right\vert ^2=\langle v_s\vert\hat{Q}_p\vert v_s\rangle$$

iar distributiile marginale sunt:

$$p(a_n)=\sum_{p=1}^Mp(a_n,b_p)=\sum_{s=1}^N\lambda _s\left\vert \alpha _{sn}\right\vert ^2$$

$$p(b_p)=\sum_{n=1}^Np(a_n,b_p)=\sum_{s=1}^N\lambda _s\left\vert \beta _{sp}\right\vert ^2$$

Utilizand expresia (3.2.1) obtinem:

$$S_{_1}(\emph{a})=-\sum_{n=1}^N\sum_{s=1}^N\lambda _s\left\vert \a... ...^2\ln \left( \sum_{r=1}^N\lambda _r\left\vert \alpha _{rn}\right\vert ^2\right)$$ (3.2.12)

Folosind proprietatea de concavitate a entropiei logaritmice, se poate arata ca (3.2.12) are un minim pentru cazul in care suma din logaritm are un singur termen, deci pentru cazul in care proiectorii $\hat{P}_n$ si $\vert u_s\rangle \langle u_s\vert$ comuta $\forall n,s$. Valoarea minima este:

$$S_{_1}(\hat{\rho}^{\prime })=-\sum_{n=1}^N\lambda _n\ln \left( \l... ... \prime }\ln \hat{\rho} ^{\prime \prime })=S_{_1}(\hat{\rho}^{\prime \prime })$$

care se numeste entropie Shannon-von Neumann a starilor reduse [234]. Aceasta marime se poate constitui intr-o masura a entanglementului. Ea este consistenta cu entropia Shannon-von Neumann pentru un operator statistic oarecare:

$$S_{_1}(\hat{\rho})=-\sum_{n=1}^N\mu _n\ln \left( \mu _n\right) =-\mathrm{tr}( \hat{\rho}\ln \hat{\rho})$$

unde $\left\{ \mu _n\right\} _n$ sunt valorile proprii (pozitive si subunitare) ale lui $\hat{\rho}$.

Putem defini atunci similar cu (3.2.3) informatia von Neumann mutuala:

$$I_1(\hat{\rho}\vert\vert\hat{\rho}^{\prime }\otimes \hat{\rho}^{\... ...})=S_1( \hat{\rho}^{\prime })+S_1(\hat{\rho}^{\prime \prime })-S_1(\hat{\rho})$$

si similar cu (3.2.2) entropia von Neumann relativa pentru 2 operatori statistici:

$$S_1(\hat{\sigma}\vert\vert\hat{\rho})=\mathrm{tr}[\hat{\sigma}(\ln \hat{\sigma}-\ln \hat{\rho})]$$ (3.2.13)

iar relatia dintre cele 2 marimi este data de:

$$S_1(\hat{\rho}\vert\vert\hat{\rho}^{\prime }\otimes \hat{\rho}^{\... ...-\ln \left( \hat{\rho}^{\prime }\otimes \hat{\rho}^{\prime \prime }\right) )]=$$

$$=\mathrm{tr}[\hat{\rho}\ln \hat{\rho}]-\mathrm{tr}[\hat{\rho}\ln ... ...\rho}\ln \left( \hat{1}^{\prime }\otimes \hat{\rho}^{\prime \prime }\right) ]=$$

$$=-S_1(\hat{\rho})-\mathrm{tr}^{\prime }[\mathrm{tr}^{\prime \prim... ..._1(\hat{\rho}\vert\vert\hat{\rho}^{\prime }\otimes \hat{\rho}^{\prime \prime })$$ (3.2.14)