9.1.1: Axiom$^{(m)}$ 1 (mixed measurement)
In the previous chapters,
we studied
Axiom 1 ( pure measurement: $\S$2.7),
that is,
In this chapter,
we will study "Axiom${}^{(m)}$ 1(mixed measurement)" in mixed measurement theory,
that is,
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$_1$)
):
In this chapter, we introduce four
"mixed measurements"
as follows.
In this book, we mainly devote ourselves to
(C$_1$)
( and sometimes
(C$_2$)
).
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,
\[
\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}
\]
$(A_1):$
$W^*$-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
$$
\mbox{
"measurement${}^{C^*}$ ${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.
$
$
S_{[{}\rho] }
\big)$"
},
\quad
\mbox{
"measurement
${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.
$
$
S_{[{}\rho] }
\big)$"
},
$$
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.
$(B_1):$
$W^*$-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
$$
\!\!\!\!\!
\mbox{
"$C^*$-mixed measurement${}^{W^*}$ ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.
$
$
{S}_{[{}\ast]}(\rho_0)
\big)$"
},
\;\;
\mbox{
"mixed measurement
${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{}}.
$
$
{S}_{[{}\ast]}(\rho_0)
\big)$"
}
$$
$(B_3):$
$W^*$-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
$$
\!\!\!\!\!
\mbox{
"$C^*$-mixed measurement${}^{C^*}$ ${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.
$
$
{\overline S}_{[{}\ast]}(\rho_0)
\big)$"
},
\;\;
\mbox{
"mixed measurement
${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{}}.
$
$
{S}_{[{}\ast]}(\rho_0)
\big)$"
}
$$
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}