## 3 Analyzing the Voting Process  
In every parliamentarian system there are many occasions where both parties and the independent legislators agree to vote in favor of a bill that, in their view, advances social welfare. In such cases, the voting process becomes a rather typical formality, as the result is predetermined in advance. Such cases will be of no concern in this paper. Instead, we shall focus on cases where the two parties and the independent representatives, if they exist, are in strong disagreement regarding a specific bill. In such situations the voting process can be viewed as a game between the two parties. We pursue this perspective in Section 5. Without loss of generality, we shall assume that [PERSON]'s party \(A\) is in favor of the bill, whereas [PERSON]'s party \(B\) is against the bill. The opposite situation can be treated in an entirely symmetric way. Figure 1 provides a simplified 2-dimensional visual representation of the situation concerning the free will radii. It can be considered a projection of the Bloch sphere onto the \(yz\) plane. Representatives from party A need not vote \(\left|yes\right\rangle\), but their vote must be within a distance of \(r_{A}\) from \(\left|yes\right\rangle\), that is on the shaded red portion of the circle. In terms of the Bloch sphere, their vote must lie on the Bloch sphere and at the same time be withing a virtual sphere centered at \(\left|yes\right\rangle\) and having radius \(r_{A}\). Symmetrically, legislators from party B may not vote \(\left|no\right\rangle\), but their vote must be within a distance of \(r_{B}\) from \(\left|no\right\rangle\), that is on the shaded green portion of the circle. In the complete 3-dimensional picture, their vote is a point on the Bloch sphere, which also happens to be inside a virtual sphere centered at \(\left|no\right\rangle\) and having radius \(r_{B}\). The representatives of [PERSON]'s party are allowed some room for maneuver, which practically means that their vote, viewed as a quantum state, need not be identical to the party recommendation, which, according to our previous assumptions, is \(\left|0\right\rangle\). Therefore, if \(\left|\psi_{j}^{A}\right\rangle\) is the quantum state corresponding to the vote of representative \(j,1\leq j\leq n_{A}\), the distance between the states \(\left|0\right\rangle\) and \(\left|\psi_{j}^{A}\right\rangle\) must be less than party A's free will radius \(r_{A}\). In a symmetrical fashion, if \(\left|\psi_{k}^{B}\right\rangle\) is the quantum state corresponding to the vote of representative \(k,1\leq k\leq n_{B}\), of [PERSON]'s party, the distance between the states \(\left|1\right\rangle\) and \(\left|\psi_{k}^{B}\right\rangle\) must be less than \(r_{B}\). A standard metric used to determine how close are two quantum states are is the _trace distance_ (see for more details the books by [18; 19]). **Definition 1** (Trace distance). : _The trace distance between two quantum states represented by their density matrices \(\rho\) and \(\sigma\) respectively, is computed by the next formula:_  
\[d_{\mathrm{tr}}(\rho,\sigma)=\frac{1}{2}\,\mathrm{tr}\,\left|\rho-\sigma \right|\text{.} \tag{3}\]  
A quick reminder is perhaps in order here: given an operator \(U\), \(U^{\dagger}\) is its _adjoint_, its _trace_\(\mathrm{tr}\,U\) is the sum of its diagonal elements, that is \(\mathrm{tr}\,U=\sum_{i}\left\langle i\left|U\right|i\right\rangle\), and \(\left|U\right|\) is defined as \(\left|U\right|=\sqrt{U^{\dagger}U}\), i.e., as the positive square root of \(U^{\dagger}U\). The general form of the density matrix \(\rho_{\psi}\) corresponding to a pure state \(\left|\psi\right\rangle=a\left|0\right\rangle+b\left|1\right\rangle\), where \(\left|a\right|^{2}+\left|b\right|^{2}=1\), is given by the next formula:  
Figure 1: This figure provides a 2-dimensional visual representation of the free will radii.