8.2: Properties of quasi-product observables

Consider the measurement ${\mathsf M}_{\overline{\mathcal A}} ( {\mathsf O}_{12} {{=}} ( X_1 \times X_2, {\cal F}_1 \boxtimes {\cal F}_2, F_{12}), S_{[\rho]})$ with the sample probability space $( X_1 \times X_2, {\cal F}_1 \boxtimes {\cal F}_2, {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\cdot ) {}\big) {}_{\overline{\mathcal A} } )$. Put

\begin{align} {\rm{Rep}}_\rho^{\Xi_1\times \Xi_2}[{}{\mathsf O}_{12}] = \left[\begin{array} {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2) {}\big) {}_{\overline{\mathcal A} } & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } \\ {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2) {}\big) {}_{\overline{\mathcal A} } & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } \end{array}\right] \quad (\forall \Xi_1 \in {\mathcal F}_1, \forall \Xi_2 \in {\mathcal F}_2) \end{align}

where, $\Xi^c$ is the complement of $\Xi$, that is, $\Xi^c=\{ x \in X \;|\; x \notin \Xi \}$. Also, note that

\begin{align} & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(1)}] (\Xi_1 ) {}\big) {}_{\overline{\mathcal A} } \\ & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(1)} (\Xi_1^c ) {}\big) {}_{\overline{\mathcal A} } \\ & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(2)} (\Xi_2^c ) {}\big) {}_{\overline{\mathcal A} } \\ & {}_{{\mathcal A}^*}\big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(2)} (\Xi_2^c ) {}\big) {}_{\overline{\mathcal A} } \end{align} We have the following lemma.
Lemma 8.4 [The condition of quasi-product observables] Consider the general basic structure \begin{align} \mbox{ {$[ {\mathcal A} \subseteq \overline{\mathcal A} \subseteq B(H)]$}. } \end{align}

Let ${\mathsf O}_{1}$ ${{=}}$ $( X_1, {\cal F}_1 , F_{1})$ and ${\mathsf O}_{2}$ ${{=}}$ $( X_2, {\cal F}_2 , F_{2}{} )$ be observables in $C (\Omega)$. Let ${\mathsf O}_{12}$ ${{=}}$ $( X_1 \times X_2, {\cal F}_1 \times {\cal F}_2, F_{12}{}{{=}} F_1 \mathop{\overset{qp}{\times}} F_2 )$ be a quasi-product observable of ${\mathsf O}_{1}$ and ${\mathsf O}_{2}$. That is, it holds that

\begin{align} F_1 = F_{12}^{(1)}, \qquad F_2= F_{12}^{(2)} \end{align}

Then, putting $\alpha_\rho^{^{\Xi_1 \times \Xi_2}} = {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2) {}\big) {}_{\overline{\mathcal A} } =\rho( F_{12} (\Xi_1 \times \Xi_2) )$, we see

\begin{align} & \; \; {\rm{Rep}}_\rho^{\Xi_1\times \Xi_2}[{}{\mathsf O}_{12}] = \left[\begin{array} {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2) {}\big) {}_{\overline{\mathcal A} } & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } \\ {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2) {}\big) {}_{\overline{\mathcal A} } & {}_{{\mathcal A}^*} \big( \rho, {}F_{12} (\Xi_1^c \times \Xi_2^c) {}\big) {}_{\overline{\mathcal A} } \end{array}\right] \nonumber \\ = & \left[\begin{array} \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho & {}\rho( {}F_1 (\Xi_1)) - \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho \\ {} \rho( {}F_2 (\Xi_2))- \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho & {} 1+ \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho - {}\rho( {}F_1 (\Xi_1)) - \rho( {}F_2 (\Xi_2)) \end{array}\right] \tag{8.2} \end{align} and \begin{align} & \max \{ 0, \rho( {}F_1 (\Xi_1))+ \rho( {}F_2 (\Xi_2)) -1 {} \} {{\; \leqq \;}} \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho {{\; \leqq \;}} \nonumber \\ & \hspace{5cm} \min \{ \rho( {}F_1 (\Xi_1)) , \; \rho( {}F_2 (\Xi_2)) \} \nonumber \\ & \hspace{4cm} (\forall \Xi_1 \in {\cal F}_1, \forall \Xi_2 \in {\cal F}_2, \forall \rho \in {\frak S}^p({\mathcal A}^*) ) \tag{8.3} \end{align}

Reversely, for any $\alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho$ satisfying (8.3), the observable ${\mathsf O}_{12}$ defined by (8.2) is a quasi-product observable of ${\mathsf O}_1$ and ${\mathsf O}_2$. Also, it holds that

\begin{align} \rho( {}F {}( \Xi_{1} \times \Xi^c_{2}) {}{} ) =0 \; & \Longleftrightarrow \; \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho = \rho( {}F_1 ( \Xi_{1} {}) ) \nonumber \\ &\Longrightarrow \; \rho( {}F_1 ( \Xi_{1} {}) ) {{\; \leqq \;}} \rho( {}F_2 ( \Xi_{2} {}) ) \tag{8.4} \end{align}

Proof. Though this lemma is easy, we add a brief proof for completeness. $0 {{\; \leqq \;}}$ $\rho( {}F (( \Xi'_1 \times \Xi'_2) {}) )$ ${{\; \leqq \;}}1$, $(\forall \Xi'_1 \in {\cal F}_1, \Xi'_2 \in {\cal F}_2 )$ we see, by (8.2) that

\begin{align} & 0 {{\; \leqq \;}}\alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho {{\; \leqq \;}}1 \\ & 0 {{\; \leqq \;}} 1+ \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho - \rho( {}F_1 ( \Xi_{1} {}) ) - \rho( {}F_2 ( \Xi_{2} {}) ) {{\; \leqq \;}}1 \\ & 0 {{\; \leqq \;}} \rho( {}F_2 ( \Xi_{2} {}) ) - \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho {{\; \leqq \;}}1 \\ & 0 {{\; \leqq \;}} \rho( {}F_1 ( \Xi_{1} {}) ) - \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho {{\; \leqq \;}}1 \end{align}

which clearly implies (8.3). Conversely. if $\alpha$ satisfies (8.3), then we easily see (8.2), Also, (8.4) is obvious. This completes the proof.

Let ${\mathsf O}_{12}$ ${{=}}$ $( X_1 \times X_2, {\cal F}_1 \boxtimes {\cal F}_2, F_{12}{}{{=}} F_1 {\mathop{\overset{qp}{\times}}_{}} F_2)$ be a quasi-product observable of ${\mathsf O}_{1}$ ${{=}}$ $( X_1, {\cal F}_1 , F_{1})$ and ${\mathsf O}_{2}$ ${{=}}$ $( X_2, {\cal F}_2 , F_{2})$ in $\overline{\mathcal A}$. Consider the measurement ${\mathsf M}_{\overline{\mathcal A}} ({\mathsf O}_{12}$ ${{=}}(X_1 \times X_2, {\cal F}_1 \boxtimes {\cal F}_2, F_{12}{}{{=}} F_1 {\mathop{\overset{qp}{\times}}} F_2),$ $S_{[\rho]})$). And assume that a measured value$(x_1, x_2)$ $(\in X_1 \times X_2 )$ is obtained. And assume that we know that $x_1 \in \Xi_1$. Then, the probability (i.e., the conditional probability) that $x_2 \in \Xi_2$ is given by

\begin{align} P= \frac{ \rho (F_{12} (\Xi_1 \times \Xi_2)) } { \rho (F_{1} (\Xi_1 )) } = \frac{ \rho (F_{12} (\Xi_1 \times \Xi_2)) } { \rho (F_{12} (\Xi_1 \times \Xi_2 )) + \rho (F_{12} (\Xi_1 \times \Xi_2^c)) } \end{align} And further, it is, by (8.3), estimated as follows. \begin{align} & \frac{\max \{ 0, \rho (F_1 ( \Xi_{1} {}) ) + \rho (F_2 ( \Xi_{2} {}) ) -1 {} \} }{{ \rho (F_{12} (\Xi_1 \times \Xi_2)) + \rho (F_{12} (\Xi_1 \times \Xi^c_2 )) }} {{\; \leqq \;}} P {{\; \leqq \;}} \\ & \hspace{4cm} \frac{ \min \{ \rho (F_1 ( \Xi_{1} {}) ) , \; \rho (F_2 ( \Xi_{2} {}) ) \} }{{ \rho (F_{12} (\Xi_1 \times \Xi_2)) + \rho (F_{12} (\Xi_1 \times \Xi_2^c)) }} \end{align}

Example 8.5 [Example of tomatoes] Let $\Omega$ $=$ $\{ \omega_1 , \omega_2 , ...., \omega_N \}$ be a set of tomatoes, which is regarded as a compact Hausdorff space with the discrete topology. Consider the classical basic structure

\begin{align} [C_0(\Omega ) \subseteq L^\infty ( \Omega, \nu ) \subseteq B(L^2 ( \Omega, \nu ))] \end{align}

Consider yes-no observables ${\mathsf O}_{RD}$ $\equiv$ $(X_{ RD} , 2^{ X_{RD} } , F_{RD})$ and ${\mathsf O}_{SW}$ $\equiv$ $(X_{SW} , 2^{ X_{SW} } , F_{SW})$ in $C(\Omega )$ such that:

\begin{align} X_{RD} = \{ y_{{}_{RD}} , n_{{}_{RD}} \} \mbox{ and } X_{SW} = \{ y_{{}_{SW}} , n_{SW} \}, \end{align}

where we consider that "$y_{{}_{RD}}$" and "$n_{{}_{RD}}$" respectively mean "RED" and "NOT RED". Similarly, "$y_{{}_{SW}}$" and "$n_{{}_{SW}}$" respectively mean "SWEET" and "NOT SWEET".

For example, the $\omega_1$ is red and not sweet, the $\omega_2$ is red and sweet, etc. as follows.

Next, consider the quasi-product observable as follows.

\begin{align} {\mathsf O}_{12} = (X_{{}_{RD}} \times X_{{}_{SW}} , 2^{ X_{{}_{RD}} \times X_{{}_{SW}} }, F {{=}} F_{{}_{RD}} \times F_{{}_{SW}}) \end{align} That is, \begin{align} & \; \; \mbox{Rep}^{\{( y_{{{}_{RD}}} , y_{{{}_{SW}}}) \}}_{\omega_k} [{\mathsf O}_{12}] = \left[\begin{array}{ll} [{}F (\{( y_{{{}_{RD}}} , y_{{{}_{SW}}}) \}) {}] ({\omega_k}) & [{}F (\{( y_{{{}_{RD}}} , n_{{{}_{SW}}}) \}) {}] ({\omega_k}) \\ {}[{}F (\{( n_{{{}_{RD}}} , y_{{{}_{SW}}}) \}) {}] ({\omega_k}) & [{}F (\{( n_{{{}_{RD}}} , n_{{{}_{SW}}}) \}) {}] ({\omega_k}) \\ \end{array}\right] \\ = & \left[\begin{array}{ll} \alpha_{_{\{( y_{{}_{RD}} , y_{{{}_{SW}}}) \} }} & [{}F_{{}_{RD}} (\{ y_{{{}_{RD}}} \}) {}] - \alpha_{_{\{( y_{{{}_{RD}}} , y_{{{}_{SW}}}) \} }} \\ {}[{}F_{{}_{SW}} (\{ y_{{{}_{SW}}} \}) {}] - \alpha_{_{\{( y_{{{}_{RD}}} , y_{{{}_{SW}}}) \} }} & 1+ \alpha_{_{\{( y_{{{}_{RD}}} , y_{{{}_{SW}}}) \} }} - [{}F_{{}_{RD}} (\{ y_{{{}_{RD}}} \}) {}] - [{}F_{{}_{SW}} (\{ y_{{{}_{SW}}} \}) {}] \\ \end{array}\right] \end{align}

where $\alpha_{_{\{( y_{{{}_{RD}}} , y_{{{}_{SW}}}) \} }} ({\omega_k})$ satisfies the (8.3). When we know that a tomato ${\omega_k}$ is red, the probability $P$ that the tomato ${\omega_k}$ is sweet is given by

\begin{align} P= \frac{ [{}F (\{(y_{{}_{RD}} , y_{{}_{SW}}) \}){}]({\omega_k}) } { [{}F (\{(y_{{}_{RD}} , y_{{}_{SW}}) \}){}]({\omega_k}) + [{}F (\{(y_{{}_{RD}} , n_{{}_{SW}}) \}){}]({\omega_k}) } = \frac{ [{}F (\{(y_{{}_{RD}} , y_{{}_{SW}}) \}){}]({\omega_k}) } { [{}F_{{}_{RD}} (\{ y_{{{}_{RD}}} \}) {}] ({\omega_k}) } \end{align}

Since $[{}F (\{(y_{{}_{RD}} , y_{{}_{SW}}) \}){}]({\omega_k})= \alpha_{_{\{(y_{{}_{RD}} , y_{{}_{SW}}) \}} } (\omega_k)$, the conditional probability $P$ is estimated by

\begin{align} & \frac{\max \{ 0, [{}F_1 ( \{ y_{{{}_{RD}}} \} {}) {}] ({\omega_k}) + [{}F_2 ( \{ y_{{{}_{SW}}} \} {}) {}] ({\omega_k}) -1 {} \} }{{ [{}F_{{}_{RD}} (\{ y_{{{}_{RD}}} \}) {}] ({\omega_k}) }} {{\; \leqq \;}} P {{\; \leqq \;}} \frac{ \min [{}F_1 ( \{ y_{{{}_{SW}}} \} {}) {}] ({\omega_k}) , \; [{}F_2 ( \{ y_{{{}_{SW}}} \} {}) {}] ({\omega_k}) \} }{{ [{}F_{{}_{RD}} (\{ y_{{{}_{RD}}} \}) {}] ({\omega_k}) }} \end{align}