4.3.3: Without the average value coincidence condition
Now we have the complete form of Heisenberg's uncertainty relation as Theorem 4.15, To be compared with Theorem 4.15, we should note that the conventional Heisenberg's uncertainty relation (= Proposition 4.10) is ambiguous. Wrong conclusions are sometimes derived from the ambiguous statement (= Proposition 4.10). For example, in some books of physics, it is concluded that EPR-experiment, or, see the following section conflicts with Heisenberg's uncertainty relation. That is,

${\rm [I]:}$ Heisenberg's uncertainty relation says that the position and the momentum of a particle can not be measured simultaneously and exactly.
On the other hand, some may consider that
${\rm [II]:}$ EPR-experiment says that the position and the momentum of a certain " particle" can be measured simultaneously and exactly ( Also, see Note4.3.)

Thus someone may conclude that the above [I] and [II] includes a paradox, and therefore, EPR-experiment is in contradiction with Heisenberg's uncertainty relation. Of course, this is a misunderstanding. Now we shall explain the solution of the paradox.

[Concerning the above [I]] Put $H= L^2 ({\mathbb R}_{q})$. Consider two-particles system in $H \otimes H = L^2 ({\mathbb R}^2_{(q_1 , q_2{})})$. In the EPR problem, we, for example, consider the state $u_e$ $({}\in H \otimes H = L^2 ({\mathbb R}^2_{(q_1 , q_2{})}))$ $\Big($or precisely, $| u_e \rangle \langle u_e | \Big)$ such that:

\begin{align} u_e ({}q_1 , q_2{}) = \sqrt{ \frac{1}{{{ 2 \pi \epsilon \sigma} }}} e^{ - \frac{1}{8 \sigma^2 } ({}{q_1 - q_2} - {a} {})^2 - \frac{1}{8 \epsilon^2 } ({}{q_1 + q_2} - {b} {})^2 } \cdot e^{ i \phi({}q_1 , q_2{}) } \tag{4.36} \end{align}

where $\epsilon$ is assumed to be a sufficiently small positive number and $\phi(q_1 , q_2{})$ is a real-valued function. Let $A_1{}\! : L^2 ({\mathbb R}^2_{(q_1 , q_2{})}) \to $ $L^2 ({\mathbb R}^2_{(q_1 , q_2{})}) $ and $A_2 \!: L^2 ({\mathbb R}^2_{(q_1 , q_2{})}) \to $ $L^2 ({\mathbb R}^2_{(q_1 , q_2{})})$ be (unbounded) self-adjoint operators such that

\begin{align} A_1 = q_1 , \qquad A_2 = \frac{ \hbar \partial }{ i \partial q_1 }. \tag{4.37} \end{align}

Then, Theorem 4.15 says that there exists an approximately simultaneous observable $(K, s, \widehat{A}_1, \widehat{A}_2)$ of $A_1$ and $A_2$. And thus, the following Heisenberg's uncertainty relation (= Theorem 4.15) holds,

\begin{align} \| {\widehat A}_1 {u_e} - A_1 {u_e} \| \cdot \| {\widehat A}_2 {u_e} - A_2 {u_e} \| \geq \hbar / 2 \tag{4.38} \end{align}

[Concerning the above [II]], However, it should be noted that, in the above situation we assume that the state $u_e$ is known before the measurement. In such a case, we may take another measurement as follows: Put $K={\mathbb C}$, $s=1$. Thus, $(H \otimes H ) \otimes K= H \otimes H$, $u \otimes s=u \otimes 1 = u $. Define the self-adjoint operators ${\widehat A}_1{}: L^2 ({\mathbb R}^2_{(q_1 , q_2{})}) \to $ $L^2 ({\mathbb R}^2_{(q_1 , q_2{})}) $ and ${\widehat A}_2: L^2 ({\mathbb R}^2_{(q_1 , q_2{})}) \to $ $L^2 ({\mathbb R}^2_{(q_1 , q_2{})})$ such that

\begin{align} {\widehat A}_1 = b - q_2, \qquad {\widehat A}_2 = A_2 = \frac{ \hbar \partial }{ i \partial q_1 } \tag{4.39} \end{align} Note that these operators commute. Therefore,
$(\sharp):$ we can take an exact simultaneous measurement of ${\widehat A}_1$ and ${\widehat A}_2$ (for the state $u_e$).
And moreover, we can easily calculate as follows: \begin{align} & \| {\widehat A}_ 1 {u_e} -A_1 {u_e} \| \nonumber \\ = & \Big[ \iint_{{\mathbb R}^2} \Big| (({}b- q_2{}) - q_1{}) \sqrt{ \frac{1}{{{ 2 \pi \epsilon \sigma} }}} e^{ - \frac{1}{8 \sigma^2 } ({}{q_1 - q_2} - {a} {})^2 - \frac{1}{8 \epsilon^2 } ({}{q_1 + q_2} - {b} {})^2 } \cdot e^{ i \phi({}q_1 , q_2{}) } \Big|^2 dq_1 dq_2 \Big]^{1/2} \nonumber \\ = & \Big[ \iint_{{\mathbb R}^2} \Big| (({}b- q_2{}) - q_1{}) \sqrt{ \frac{1}{{{ 2 \pi \epsilon \sigma} }}} e^{ - \frac{1}{8 \sigma^2 } ({}{q_1 - q_2} - {a} {})^2 - \frac{1}{8 \epsilon^2 } ({}{q_1 + q_2} - {b} {})^2 } \Big|^2 dq_1 dq_2 \Big]^{1/2} \nonumber \\ = & {\sqrt 2} \epsilon , \tag{4.40} \end{align} \begin{align} & \| {\widehat A}_2 {u_e} - A_2 {u_e} \| = 0. \tag{4.41} \end{align} Thus we see \begin{align} \| {\widehat A}_1 {u_e} - A_1 {u_e} \| \cdot \| {\widehat A}_2 {u_e} - A_2 {u_e} \| = 0. \tag{4.42} \end{align} However it should be again noted that, the measurement $({}\sharp{})$ is made from the knowledge of the state $u_e$.

[[I] and [II] are consistent] The above conclusion (4.43) does not contradict Heisenberg's uncertainty relation (4.38), since the measurement $(\sharp)$ is not an approximate simultaneous measurement of $A_1$ and $A_2$. In other words, the $(K,s, \widehat{A}_1, \widehat{A}_2 )$ is not an approximately simultaneous observable of $A_1$ and $A_2$. Therefore, we can conclude that

$(F):$Heisenberg's uncertainty principle is violated without the average value coincidence condition
$\fbox{Note 4.3}$ Some may consider that the formulas (4.40) and (4.41) imply that the statement [II] is true. However, it is not true. This is answered in Remark 8.15.
Also, we add the following remark.
Remark 4.16 Calculating the second term (precisely , $\langle u \otimes s,$"the second term"$(u \otimes s) \rangle$) and the third term (precisely , $\langle u \otimes s,$"the third term"$(u \otimes s) \rangle$) in (4.25), we get, by Robertson's uncertainty principle (4.19), \begin{align} & 2 {\overline \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot \sigma(A_2;u) \ge |\langle u \otimes s, [{\widehat N}_1, A_2 \otimes I](u \otimes s) \rangle| \tag{4.43} \\ & 2 {\overline \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} \cdot \sigma(A_1;u) \ge |\langle u \otimes s, [A_ \otimes I, {\widehat N}_2](u \otimes s) \rangle | \tag{4.44} \\ & \qquad ( \forall u \in H \mbox{ such that } ||u||=1) \nonumber \end{align} and, from (4.25),(4.26), (4.43),(4.44), we can get the following inequality \begin{align} & { \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot { \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} +{ \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} \cdot \sigma(A_1;u) +{ \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot \sigma(A_2;u) \nonumber \\ \ge & {\overline \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot {\overline \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} +{\overline \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} \cdot \sigma(A_1;u) +{\overline \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot \sigma(A_2;u) \nonumber \\ \ge & \frac{1}{2} | \langle u , [A_1,A_2] u \rangle | \quad ( \forall u \in H \mbox{ such that } ||u||=1) \tag{4.45} \end{align} Since we do not assume the average value coincidence condition, it is a matter of course that this (4.45) is more rough than Heisenberg's uncertainty principle (4.34)

Under a certain interpretation such that ${ \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} $ and ${ \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} $ can be respectively regarded as "error $\epsilon (A_1,u)$" and "disturbance :$\eta (A_2,u)$", the iequality (4.45), that is, $$ \epsilon(A_1, u ) \eta( A_2, u ) + \epsilon(A_1, u ) \sigma( A_2, u ) + \sigma( A_1, u ) \eta( A_2, u ) \ge \frac{1}{2} | \langle u , [A_1,A_2] u \rangle | $$ is called Ozawa's inequality.
However, since the linguistic interpretation says that

\begin{align*} ---\mbox{ Only one measurement is permitted. And thus, we say nothing after measurement. } --- \end{align*}

the term "disturbance" can not be used in the linguistic interpretation (cf. S. Ishikawa, arXiv:1308.5469 [quant-ph] 2014 ).