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}