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. |

${\rm [II]:}$ | 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 $u_e$). |

[[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. |

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

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