# Bell-Clauser-Horne-Shimony-Holt Behavior Under Quantum Loss and Decoherence
## 2 Background, Notation
### Two-Party Bell Experiments as Channels  
The bipartite joint probabilities, \(P(ab|xy)\), are crucial objects for the study of quantum nonlocality and no-signaling theories. Consider a bipartite Bell-type experiment. Using the previous metaphor of actors, [PERSON] and [PERSON] are now at the same location, whereas [PERSON] and [PERSON] are together at another, separated place. So, the parties will be referred to as [PERSON] and [PERSON]. [PERSON] and [PERSON] share a pair of quantum systems in an entangled state. [PERSON] (and [PERSON]) can choose from a set of quantum measurements, \(X\). Upon carrying out the selected measurement, \(x\), they obtain a result \(a\in A\). Similarly, [PERSON] (and [PERSON]) choose a measurement, \(y\in Y\), and obtain a result, \(b\in B\). Given the set of measurements and the entangled state, according to the rules of quantum mechanics, the conditional probability distribution, \(P(ab|xy)\), can be calculated, which is commonly termed as \"behavior\" in the present context. A behavior can exhibit so-called nonclassical correlations between [PERSON] and [PERSON]. The peculiarity of such correlations lies in the fact that [PERSON] + [PERSON] are separated from [PERSON], so their channels do not interact. Yet, there can be correlations that cannot be explained, even assuming the pre-shared randomness (i.e., the keys) between the two locations. The Bell inequalities are inequalities that can be expressed in terms of \(P(ab|xy)\) and their violation indicates this unexplainability. This scenario can be extended to study general behaviors, \(P(ab|xy)\). The no-signaling ones, that is, those that do not facilitate communication between Alice+Xenia and [PERSON]+Yvette, are of special interest. Formally, the no-signaling constraints in our notation can be written as  
\[\sum_{b}T^{(XY\to AB)}_{xy,ab} =\sum_{b}T^{(XY\to AB)}_{xy^{\prime},ab}\text{ for all }x,a,y,y^{\prime} \tag{7}\] \[\sum_{a}T^{(XY\to AB)}_{xy,ab} =\sum_{a}T^{(XY\to AB)}_{x^{\prime}y,ab}\text{ for all }y,b,x,x^{\prime}. \tag{8}\]  
These constraints imply that the local marginal probabilities of Alice can be defined unambiguously as  
\[T^{(X\to A)}_{x,a}:=T^{(XY\to AB)}_{xy,a}=\sum_{b}T^{(XY\to AB)}_{xy,ab}, \tag{9}\]  
as these values are independent of [PERSON]'s measurement setting, \(y\) (and the other way around). These form a polytope in the vector space of the behaviors, \(T^{(XY\to AB)}\). Note that, from the classical channel's point of view, in the Bell scenario, there are two sender-receiver pairs [PERSON]\(-\)[PERSON] and [PERSON]\(-\)[PERSON]. The members of each of these pairsare, however, located at the same place. So, the correlations are not between the \"sender\" and the \"receiver\" in the channel sense, but between the two parts of the channel. Yet, in some cases, it is useful to view them from the channel's perspective, writing the behavior \(P(ab|xy)\) in the form of a channel matrix, \(T^{(XY\to AB)}\). As an example, consider the celebrated Clauser-Horne-Shimony-Holt (CHSH) experiment, which will be the subject of the present study. [PERSON] and [PERSON] share a pair of quantum bits in a maximally entangled state; let it be the Bell state:  
\[\ket{\Phi^{(+)}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}), \tag{10}\]  
where we have fixed a product basis, \(\ket{i,j}\), \(\left(i,j\right)\in\{0,1\}^{2}\), on the space of the two qubits; the first index is for [PERSON]'s side. [PERSON]'s measurement bases are:  
\[\ket{a=0}_{x=0}=\ket{0},\quad\ket{a=1}_{x=0}=\ket{1},\] \[\ket{a=0}_{x=1}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1}),\quad\ket{a=1 }_{x=1}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1}), \tag{11}\]  
whereas [PERSON]'s ones are:  
\[\ket{b=0}_{y=0}=\cos\theta\ket{0}+\sin\theta\ket{1},\quad\ket{b=1 }_{y=0}=-\sin\theta\ket{0}+\cos\theta\ket{1},\] \[\ket{b=0}_{y=1}=\cos\theta\ket{0}-\sin\theta\ket{1},\quad\ket{b= 1}_{y=1}=\sin\theta\ket{0}+\cos\theta\ket{1}, \tag{12}\]  
where \(\theta=\pi/8\).