## 4 Observables
### Our Approach
: _The \(\tau\)-observed computation of a QTM \(M\) on the initial \(q\)-configuration \(|\phi\rangle\), is the set \(\mathcal{K}^{M}_{|\phi\rangle,\tau}\) of the \(\tau\)-observed runs of \(M\) on \(|\phi\rangle\) with the measure \(\mathsf{Pr}:\mathcal{P}(\mathcal{K}^{M}_{|\phi\rangle,\tau})\to\mathbb{C}\) (where \(\mathcal{P}(\mathcal{K}^{M}_{|\phi\rangle,\tau})\) is the set of the subsets of \(\mathcal{K}^{M}_{|\phi\rangle,\tau}\)) defined by_  
\[\mathsf{Pr}\,\mathcal{B}=\sum_{R\in\mathcal{B}}\mathsf{Pr}\{R\}\]  
_for every \(\mathcal{B}\subseteq\mathcal{K}^{M}_{|\phi\rangle,\tau}\)._By \(\mathcal{K}[k]^{M}_{|\phi\rangle,\tau}\) we shall denote the set of the finite \(\tau\)-observed runs of length \(k\) of \(M\) on \(|\phi\rangle\), with the measure \(\mathsf{Pr}\) on its subsets (see Definition 21). It is immediate to observe that the set \(\mathcal{K}^{M}_{|\phi\rangle,\tau}\) naturally defines an infinite tree labelled with q-configurations, where each infinite path starting from the root \(|\phi\rangle\) corresponds to a \(\tau\)-observed run in \(\mathcal{K}^{M}_{|\phi\rangle,\tau}\). Lemma 1: _Given \(R_{1},R_{2}\in\mathcal{K}^{M}_{|\phi\rangle,\tau}\) with \(R_{1}=\{\phi_{1,i}\}_{i\in\mathbb{N}}\
eq\{\phi_{2,i}\}_{i\in\mathbb{N}}=R_{2}\), there is \(k\geq 0\) s.t._  
1. \(\phi_{1,i}=\phi_{2,i}\) _for_ \(i\leq\tau(k)\)_, that is,_ \(R_{1}[\tau(k)]=R_{2}[\tau(k)]\)_;_
2. _for_ \(i>\tau(k)\)_, the q-configurations_ \(\phi_{1,i}\
eq\phi_{2,i}\) _are in two orthonormal subspaces generated by two distinct subsets of_ \(\mathfrak{C}\)_._  
Proof: Let \(R_{1}[h]=R_{2}[h]\) be the longest common prefix of \(R_{1}\) and \(R_{2}\). Because they both starts with \(|\phi\rangle\), such a prefix is not empty; moreover, by the definition of \(\tau\)-observed run, it is readily seen that \(h=\tau(k)\), for some \(k\). Let us now prove item 2 of Lemma 1. If we take \(\mathcal{B}_{a,j}=\{C\in\mathfrak{C}\mid|C\rangle\in|\psi_{a,h+1+j}\rangle\}\), for \(a=1,2\) and \(j\in\mathbb{N}\), we need to prove that \(\mathcal{B}_{1,j}\cap\mathcal{B}_{2,j}=\emptyset\), for every \(j\). By construction, \(\phi_{1,h+1}\
eq\phi_{2,h+1}\) and, for \(a=1,2\), \(\phi_{h}\downarrow_{\chi_{a}}\phi_{a,h+1}\), for some \(x_{a}\), with \(\phi_{h}=\phi_{a,h}\). Moreover, \(\mathcal{B}_{1,0}\cap\mathcal{B}_{2,0}=\emptyset\) (see Remark 8), at least one of the two observed values \(x_{1},x_{2}\) is not \(\perp\), and one of the two q-configurations \(|\phi_{1,h+1}\rangle,|\phi_{2,h+1}\rangle\) is a final base q-configuration. W.l.o.g., let us assume that \(x_{1}\in\mathbb{N}\) and let \(|\phi_{1,h+1}\rangle=e|C,n\rangle\), for some final plain configuration \(C\) and some \(n\in\mathbb{N}\). Subsequently, \(|\phi_{1,h+1+j}\rangle=e\,U^{j}|C,n\rangle=e|C,n+j\rangle\) (see Remark 9) and \(\mathcal{B}_{1,j}=\{|C,n+j\rangle\}\), for \(j\in\mathbb{N}\). Thus, to prove the assertion, it suffices to show that \(|C,n+j\rangle\
ot\in\mathcal{B}_{2,j}\), for \(j\in\mathbb{N}\). Let us distinguish two cases:  
1. \(x_{2}\in\mathbb{N}\) and \(|\psi_{2,h+1}\rangle=e^{\prime}|D,m\rangle\), where \(D\) is a final plain configuration and \(m\in\mathbb{N}\). As already seen for \(|\psi_{1,h+1+j}\rangle\), for \(j\in\mathbb{N}\), we have \(|\psi_{2,h+1+j}\rangle=e^{\prime}\,U^{j}|D,m\rangle=e^{\prime}|D,m+j\rangle\) and \(\mathcal{B}_{2,j}=\{|D,m+j\rangle\}\). Therefore, \(|C,n+j\rangle\
ot\in\mathcal{B}_{2,j}\), since \(|C,n+j\rangle\
eq|D,m+j\rangle\), for \(|C,n\rangle\
eq|D,m\rangle\) by construction. 2. \(|\psi_{2,h+1}\rangle\in\ell^{2}(\mathfrak{C}\setminus\mathfrak{C}^{fin})\). Let \(|D,m\rangle\in\mathcal{B}_{2,j}\cap\mathfrak{C}^{fin}\), where \(D\) is a plain configuration and \(m\in\mathbb{N}\). By induction on \(j\), it is readily seen that \(m<j\leq n+j\). Thus, \(|D,m\rangle\
eq|C,n+j\rangle\).