## 4 Observables
### Our Approach
\(\quad\mathsf{P}_{|\phi_{k}\rangle}(n)=\mathsf{Pr}\{R\in\mathcal{X}[k]_{|\phi \rangle,\tau}^{M}\mid R\downarrow_{n}\}\)_, for_ \(k=\tau(i)+1\) _and_ \(i\in\mathbb{N}\)_; and,_
2. \(\quad\mathsf{P}_{K_{|\phi\rangle}^{M}}(n)=\mathsf{Pr}\{R\in\mathcal{X}_{|\phi \rangle,\tau}^{M}\mid R\downarrow_{n}\}\)_._  
Let us start with the first item. By Lemma 2, we know that  
\[\mathsf{Pr}\,\mathcal{B}=\sum_{R\in\mathcal{B}}\mathsf{Pr}\{R\}\]1where \(|\psi_{R}\rangle\) is the last q-configuration of \(R\). Because \(k=\tau(i)+1\), for some \(i\), we also know that either \(\psi_{R}\in\mathfrak{C}\setminus\mathfrak{C}^{fin}\) or \(|\psi_{R}\rangle=u_{R}|C_{R}\rangle\) with \(C_{R}\in\mathfrak{C}^{fin}\) and \(|u_{R}|=1\). Therefore,  
\[\mathsf{Pr}\,\mathcal{B}=\sum_{R\in\mathcal{B}}\mathsf{Pr}\{R\}\]2  
where  
\[\mathsf{Pr}\,\mathcal{B}=\sum_{R\in\mathcal{B}}\mathsf{Pr}\{R\}\]3 \[\mathsf{Pr}\,\mathcal{B}=\sum_{R\in\mathcal{B}}\mathsf{Pr}\{R\}\]4  
By Lemma 1, we know that for every \(R_{1},R_{2}\in\mathcal{B}[k,n]\), we have \(|C_{R_{1}}\rangle\
eq|C_{R_{2}}\rangle\). Therefore  
\[\mathsf{Pr}\,\mathcal{B}=\sum_{R\in\mathcal{B}}\mathsf{Pr}\{R\}\]5  
since \(|u_{R}|=1\). Which concludes the proof of the first item of the assertion. In order to prove the second item, let \(\mathcal{B}[\omega,n]=\{R\in\mathcal{X}_{|\phi\rangle,\tau}^{M}\mid R\downarrow_ {n}\}\). We have that  
\[\mathsf{Pr}\{R\in\mathcal{X}_{|\phi\rangle,\tau}^{M}\mid R \downarrow_{n}\} = \sum_{R\in\mathcal{B}[\omega,n]}\mathsf{Pr}\{R\}=\lim_{i\to \infty}\sum_{ \begin{subarray}{c}R\in\mathcal{B}[\omega,n]\\ R[\tau(i)+1]\downarrow_{n}\end{subarray} }\mathsf{Pr}\{R\}\] \[{}^{(*)} = \lim_{i\to\infty}\sum_{ \begin{subarray}{c}R\in\mathcal{B}[\omega, n]\\ R[\tau(i)+1]\downarrow_{n}\end{subarray} }\mathsf{Pr}\{R[\tau(i)+1]\}\] \[{}^{(**)} = \lim_{i\to\infty}\sum_{R^{\prime}\in\mathcal{B}[\tau(i)+1,n]} \mathsf{Pr}\{R^{\prime}\}=\lim_{i\to\infty}\mathsf{Pr}\,\mathcal{B}[\tau(i)+1,n]\]  
since: \((*)\mathsf{Pr}\{R\}=\mathsf{Pr}\{R[\tau(i)+1]\}\), when \(R[\tau(i)+1]\downarrow_{n}\) (see Remark 10); \((**)\), there is a bijection between \(\mathcal{B}[\tau(i)+1,n]\) and \(\{R\in\mathcal{B}[\omega,n]\mid R[\tau(i)+1]\downarrow_{n}\}\) (see Remark 9) mapping every \(R^{\prime}\in\mathcal{B}[\tau(i)+1,n]\) with last q-configuration \(u|C\rangle\) into \(R=\{\psi_{j}\}_{j\in\mathbb{N}}\in\{R\in\mathcal{B}[\omega,n]\mid R[\tau(i)+1 ]\downarrow_{n}\}\) s.t \(R^{\prime}=R[\tau(i)+1]\) and \(|\psi_{j}\rangle=u\,U_{M}^{j-\tau(i)-1}|C\rangle\) for \(j>\tau(i)\). Therefore, by the (already proved) first item of the assertion  
\[\mathsf{Pr}\{R\in\mathcal{X}_{|\phi\rangle,\tau}^{M}\mid R\downarrow_{n}\}= \lim_{i\to\infty}\mathsf{Pr}\,\mathcal{B}[\tau(i)+1,n]=\lim_{i\to\infty} \mathbf{P}_{|\phi_{\tau(i)+1}\rangle}(n)=\mathbf{P}_{K_{|\phi\rangle}^{M}}(n)\]  
The above theorem shows that our protocol is not only realistic (unlike those of B&V and Deutsch), but also compatible with the theoretical probabilities defined for configurations in superposition. In particular, the theorem shows how it is possible to make measurements in a totally free way, without destroying the expressive power of quantum parallelism, i.e., without destroying the ability of a QTM to use quantum interference.