# Bell-Clauser-Horne-Shimony-Holt Behavior Under Quantum Loss and Decoherence
## 5 Decomposability of Behaviors  
The question naturally arises whether the impact of the quantum channels, as modeled by the CPTP maps, can be replaced on the behavior's level by the composition of the ideal behavior with an effective noise channel. In other words, we ask whether the behavior \(T^{(XY\to RS)}\) obtained in the CHSH experiment--performing the original ideal measurement after the entangled state--was subjected to the effect of a pair of quantum noise channels, and whether it can be written as a composition of the ideal \(T^{(XY\to AB)}\) given in Equation (15) and an effective classical noise channel \(T^{(AB\to RS)}\) as follows:  
\[T^{(XY\to RS)}=T^{(XY\to AB)}T^{(AB\to RS)}. \tag{50}\]  
This seems plausible, and because of the CPTP maps are independent, one may even expect \(T^{(AB\to RS)}\) to be in a product form, as in the case of modeling detector inefficiencies in Equation (17). Somewhat surprisingly, at least in case of the studied CPTP map pairs, it is not the case. Mathematically, we are searching for \(T^{(AB\to RS)}\), obeying Equation (50), while the other two matrices are fixed. In addition, \(T^{(AB\to RS)}\) has to be a stochastic matrix as it represents a channel. Let us postpone, for the time being, the check of the stochasticity and investigate other necessary conditions for any \(T^{(AB\to RS)}\) obeying Equation (50) to exist. The problem of finding \(T^{(AB\to RS)}\) is a matrix inversion. However, \(T^{(XY\to AB)}\) is singular,  
Figure 2: The left-hand side of CHSH inequality as a function of quantum channel parameters. (**Left**): phase damping at Alice, amplitude damping at Bob. (**Right**): amplitude damping at Alice, phase damping at Bob. and so is \(T^{(XY\to RS)}\), at least for the studied CPTP map pairs. Hence, a necessary condition for the desired \(T^{(AB\to RS)}\) to exist is that the left null space of the \"ideal\" channel, \(T^{(XY\to AB)}\), is contained in the left null space of the \"observed\" channel, \(T^{(XY\to RS)}\). The eigenvalues of \(T^{(XY\to AB)}\) are \(1,\beta-\alpha,0,0\) and the corresponding right eigenvectors read:  
\[ \begin{pmatrix}1\\ 1\\ 1\\ 1\end{pmatrix} ^{\prime} \begin{pmatrix}-\alpha\\ -\alpha\\ 2\alpha+\beta\end{pmatrix} ^{\prime} \begin{pmatrix}1\\ 0\\ 0\\ -1\end{pmatrix} , \begin{pmatrix}0\\ 1\\ -1\\ 0\end{pmatrix} , \tag{51}\]  
where the parameters \(\alpha=c^{2}/2\) and \(\beta=s^{2}/2\) are the same as in Equation (15). The left null space is orthogonal to those right eigenvectors that correspond to nonzero eigenvalues. Hence, this is a rank-2 matrix. Let us now briefly consider the marginals defined in Equation (9); these are the behaviors \(T^{(X\to R)}\) and \(T^{(Y\to S)}\), observable on each individual side. Note that these could be obtained by applying the quantum channel on the given side to the respective part of the entangled resource, and then the measurement results can be calculated locally. In the case of the Bell-CHSH situation, the entangled resource is a Bell state; thus, the subsystems are in a completely mixed state. As these local behaviors depend on the channel on the given side, they can be calculated for the two channel types, regardless of the channel on the other side. The phase-damping channel is unital, i.e., it does not alter the completely mixed state. Thus, it does not affect the local behavior; it remains  
\[T^{\text{PD, A}}=T^{\text{PD, B}}=\frac{1}{2} \begin{pmatrix}1&1\\ 1&1\end{pmatrix} . \tag{52}\]  
In the case of amplitude damping with parameter \(p\), we obtain  
\[T^{\text{AD, A}}=\frac{1}{2} \begin{pmatrix}1+p&1-p\\ 1&1\end{pmatrix} ,T^{\text{AD, B}}=\frac{1}{2} \begin{pmatrix}1+2 csp&1-2 csp\\ 1+2 csp&1-2 csp\end{pmatrix}  \tag{53}\]  
on [PERSON]'s and [PERSON]'s sides, respectively. (Recall that the measurement bases differ on the two sides.) It is interesting to note that, on [PERSON]'s side, we can decompose the local behavior \(T^{\text{AD, B}}\) into the product of the ideal local behavior and a noise channel:  
\[\frac{1}{2} \begin{pmatrix}1&1\\ 1&1\end{pmatrix} \frac{1}{2}