9.1.1: Axiom$^{(m)}$ 1 (mixed measurement)

In the previous chapters, we studied Axiom 1 ( pure measurement: $\S$2.7), that is,

$\underset{\mbox{ (=quantum language)}}{\fbox{pure measurement theory (A)}} := \underbrace{ \underset{\mbox{ ($\S$2.7)}}{ \overset{ [\mbox{ (pure) Axiom 1}] }{\fbox{pure measurement}} } + \underset{\mbox{ ( $\S$10.3)}}{ \overset{ [{\mbox{ Axiom 2}}] }{\fbox{Causality}} } }_{\mbox{ a kind of incantation (a priori judgment)}} + \underbrace{ \underset{\mbox{ ($\S$3.1) }} { \overset{ {}}{\fbox{Linguistic interpretation}} } }_{\mbox{ the manual on how to use spells}} \tag{9.1}$

In this chapter, we will study "Axiom${}^{(m)}$ 1(mixed measurement)" in mixed measurement theory, that is,

$\underset{\mbox{ (=quantum language)}}{\fbox{mixed measurement theory (A)}} := \underbrace{ \color{red}{ \underset{\mbox{ ($\S$9.1)}}{ \overset{ [\mbox{ (mixed) Axiom 1}] }{\fbox{mixed measurement}} } } + \underset{\mbox{ ( $\S$10.3)}}{ \overset{ [{\mbox{ Axiom 2}}] }{\fbox{Causality}} } }_{\mbox{ a kind of incantation (a priori judgment)}} + \underbrace{ \underset{\mbox{ ($\S$3.1) }} { \overset{ {}}{\fbox{Linguistic interpretation}} } }_{\mbox{ the manual on how to use spells}} \tag{9.2}$

Now we will propose Axiom${}^{(m)}$ 1 (mixed type) as follows.

In the previous chapters, we mainly devoted ourselves to the following (A) (paricularly, "$W^*$-measurement (A$_1$) ):

Review 9.1 [=Preparation 2.30].
 $(A_1):$ $W^*$-measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ $S_{[{}\rho] } \big)$, where${\mathsf O}{{=}}$ $(X, {\cal F} , F)$ is a $W^*$-observable in $\overline{\mathcal A}$, a pure state $\rho (\in {\frak S}^p({\mathcal A}^*))$, Here, "$W^*$-measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O},$ $S_{[{}\rho] } \big)$" is also denoted by $$\mbox{ "measurement{}^{W^*} {\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.   S_{[{}\rho] } \big)" }, \quad \mbox{ "measurement {\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.   S_{[{}\rho] } \big)" },$$ $(A_2):$ $C^*$-measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ $S_{[{}\rho] } \big)$, where ${\mathsf O}{{=}}$ $(X, {\cal F} , F)$ is a $C^*$-observable in ${\mathcal A}$, a pure state $\rho (\in {\frak S}^p({\mathcal A}^*))$, Here, "$C^*$-measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O},$ $S_{[{}\rho] } \big)$" is also denoted by $$\mbox{ "measurement{}^{C^*} {\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.   S_{[{}\rho] } \big)" }, \quad \mbox{ "measurement {\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.   S_{[{}\rho] } \big)" },$$

In this chapter, we introduce four "mixed measurements" as follows.

Preparation 9.2

 $(B_1):$ $W^*$-mixed measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ ${\overline S}_{[{}\ast]}(w_0) \big)$, where${\mathsf O}{{=}}$ $(X, {\cal F} , F)$ is a $W^*$-observable in $\overline{\mathcal A}$, a $W^*$-mixed state $w_0 (\in \overline{\frak S}^m(\overline{\mathcal A}_*))$, Here, "$W^*$-mixed measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O},$ ${\overline S}_{[{}\ast]}(w_0) \big)$" is also denoted by $$\!\!\!\!\! \mbox{ "W^*-mixed measurement{}^{W^*} {\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.   {\overline S}_{[{}\ast]}(w_0) \big)" }, \;\; \mbox{ "mixed measurement {\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.   {\overline S}_{[{}\ast]}(w_0) \big)" }$$ $(B_2):$ $C^*$-mixed measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ ${S}_{[{}\ast]}(\rho_0) \big)$, where${\mathsf O}{{=}}$ $(X, {\cal F} , F)$ is a $W^*$-observable in $\overline{\mathcal A}$, a $C^*$-mixed state $\rho_0 (\in {\frak S}^m({\mathcal A}^*))$, Here, "$C^*$-mixed measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O},$ ${S}_{[{}\ast]}(\rho_0) \big)$" is also denoted by $$\!\!\!\!\! \mbox{ "C^*-mixed measurement{}^{W^*} {\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.   {S}_{[{}\ast]}(\rho_0) \big)" }, \;\; \mbox{ "mixed measurement {\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.   {S}_{[{}\ast]}(\rho_0) \big)" }$$
We mainly devote ourselves to the above two. Also, $\overline{S}_{[{}\ast]}$ in (B$_1$) may be written by ${S}_{[{}\ast]}$, cf. Remark 9.3 later. Thus the following are not necessarily important.

 $(B_3):$ $W^*$-mixed measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ ${\overline S}_{[{}\ast]}(w_0) \big)$, where${\mathsf O}{{=}}$ $(X, {\cal F} , F)$ is a $C^*$-observable in ${\mathcal A}$, a $W^*$-mixed state $w_0 (\in \overline{\frak S}^m(\overline{\mathcal A}_*))$, Here, "$W^*$-mixed measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O},$ ${\overline S}_{[{}\ast]}(w_0) \big)$" is also denoted by $$\!\!\!\!\! \mbox{ "W^*-mixed measurement{}^{C^*} {\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.   {\overline S}_{[{}\ast]}(w_0) \big)" }, \;\; \mbox{ "mixed measurement {\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.   {\overline S}_{[{}\ast]}(w_0) \big)" }$$ $(B_4):$ $C^*$-mixed measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ ${S}_{[{}\ast]}(\rho_0) \big)$, where${\mathsf O}{{=}}$ $(X, {\cal F} , F)$ is a $C^*$-observable in ${\mathcal A}$, a $C^*$-mixed state $\rho_0 (\in {\frak S}^m({\mathcal A}^*))$, Here, "$C^*$-mixed measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O},$ ${S}_{[{}\ast]}(\rho_0) \big)$" is also denoted by $$\!\!\!\!\! \mbox{ "C^*-mixed measurement{}^{C^*} {\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.   {\overline S}_{[{}\ast]}(\rho_0) \big)" }, \;\; \mbox{ "mixed measurement {\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.   {S}_{[{}\ast]}(\rho_0) \big)" }$$

In this book, we mainly devote ourselves to (C$_1$) ( and sometimes (C$_2$) ).

(C):$\qquad$Axiom${}^{(m)}$ 1 (mixed measurement)

Let ${\mathsf O}{{=}}$ $(X, {\cal F} , F)$ be a $W^\ast$-observable in $\overline{\mathcal A}$

(C$_1$): Let $w_0 \in \overline{\frak S}^m(\overline{\mathcal A}_*)$. The probability that a measured value obtained by $W^*$-mixed measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ ${\overline S}_{[{}\ast] }(w_0) \big)$ belongs to $\Xi$ $(\in {\cal F})$ is given by \begin{align} {}_{ {\overline{\mathcal A}}_*} (w_0 , F(\Xi) )_{\overline{\mathcal A}} \;\;\; \Big( \equiv w_0 (F(\Xi)) \Big) \end{align}

(C$_2$): Let $\rho_0 \in {\frak S}^m({\mathcal A}^*)$. The probability that a measured value obtained by $C^*$-mixed measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{=}}$ $(X, {\cal F} , F),$ $S_{[{}\ast] }(\rho_0) \big)$ belongs to $\Xi$ $(\in {\cal F})$ is given by \begin{align} {}_{ {{\mathcal A}}^*} (\rho_0 , F(\Xi) )_{\overline{\mathcal A}} \;\;\; \Big( \equiv \rho (F(\Xi)) \Big) \end{align}

As we $\color{red}{\mbox{leared}}$ Axiom 1 $\color{red}{\mbox{by rote}}$ in pure measurement theory,

we have to learn Axiom${}^{(m)}$ 1 by rote, and exercise a lot of examples
The practices will be done in this chapter.

Remark 9.3 In the above Axiom${}^{(m)}$ 1, (C$_1$) and (C$_2$) are not so different.

 $(\sharp_1):$ In the quantum case, (C$_1$)=(C$_2$) clearly holds, since ${\frak S}^m({\mathcal Tr}(H))=\overline{\frak S}^m({\mathcal Tr}(H))$ in (2.17). $(\sharp_2):$ In the classical case, we see \begin{align} L^1_{+1}( \Omega. \nu ) \ni w_0 \xrightarrow[]{\rho_0(D) = \int_D w_0 (\omega ) \nu(d \omega ) } \rho_0 \in {\mathcal M}_{+1}(\Omega ) \end{align} Therefore, in this case, we consider that \begin{align} {\mathsf M}_{L^\infty ( \Omega. \nu )} \big({\mathsf O}{{=}} (X, {\cal F} , F), {\overline S}_{[{}\ast] }(w_0) \big) = {\mathsf M}_{L^\infty ( \Omega. \nu )} \big({\mathsf O}{{=}} (X, {\cal F} , F), S_{[{}\ast] }(\rho_0) \big) \end{align}

Hence, (C$_1$) and (C$_2$) are not so different. In oder to avoid the confusion, we use the following notation:

\begin{align} \left\{\begin{array}{ll} \mbox{$W^*$-mixed state $w_0$ $( \in \overline{\frak S}^m(\overline{\mathcal A}_*)$ is written by Roman alphabet (e.g., $w_0, w, v,...$)} \\ \\ \mbox{$C^*$-mixed state $\rho_0$ $( \in {\frak S}^m({\mathcal A}^*)$ is written by Greek alphabet (e.g., $\rho_0, \rho, \nu, ...$)} \end{array}\right. \end{align}