# Bell-Clauser-Horne-Shimony-Holt Behavior Under Quantum Loss and Decoherence
## 6 Conclusions and Outlook  
We have studied the nonlocal behavior of the CHSH-Bell experiment under the presence of quantum noise affecting the elements of the entangled resource separately. To gain insight into how such imperfections influence the system, we adopted a picture of classical memoryless channels. Our main contribution is this picture of the imperfect Bell experiment, and the study of the decomposability of the so-arising classical channel. Although we examined only a few basic examples, these demonstrate that the manner in which quantum nonlocality is disrupted by quantum channels differs from how detector imperfections affect such scenarios. Specifically, the modified nonlocal behavior is not equivalent to the behavior obtainable by sending ideal measurement results through a classical channel that models local noise. In the case of unital channels, the change in the local behavior can at least be viewed as a composition of ideal behavior and classical noise; however, this approach proves to be impossible for amplitude damping. This implies that, when studying the effects of imperfections--such as, e.g., on a device-independent quantum key distribution scheme--considering only the impact of local classical noise (e.g., detector inefficiencies) does not provide a complete picture. Quantum imperfections can cause changes that are not equivalent to detector imperfections; this has to be taken into account when analyzing entanglement-based protocols. Even though our contribution is mainly theoretical and aims to build deeper understanding of the nonlocal behavior in a Bell-CHSH experiment based on the picture of a classical channel between the input pair and the output pair, the introduced technique can be useful in experimental or theoretical scenarios involving two-qubit Bell experiments. One could gain detailed insights into changes in behavior due to the imperfections by carrying out the consideration described here, for the case of the actual CPTP maps describing a practical scenario. This can be determined, e.g., via quantum channel tomography [28], which can be addressed with machine learning approaches too [29]. In this way, the change in the behavior can be analyzed, and the decomposability to the ideal behavior and classical noise channel can be addressed. The additional effect of detector efficiencies can also be considered. Our consideration can be generalized to arbitrary pairs of qubit channels. For instance, the example in Ref. [25] considers a pair of identical amplitude-damping channels acting on a singlet state; this could be another example to study as the amplitude-damping process modifies the singlet state in a way different from the entangled resource considered by us. Further, by considering the Choi representation of the most general channels, we can see that, through the linear space of semidefinite matrices--with a trace condition and the property of positive semidefiniteness--a fully general description can be obtained. Relaxing the condition of positive semidefiniteness would introduce nonphysical channels, but the calculation remains feasible. The large number of parameters, however, would make the study less transparent. It would therefore be interesting to identify more useful general conditions on decomposability, and this could potentially applicable to many-input-many-output or multiparty scenarios. Conceptualization, M.K. ; methodology, M.K., A.B., and O.H. ; software, O.H. ; validation, O.H., A.B., and P.A. ; formal analysis, O.H. and A.B. ; investigation, O.H. ; writing--original draft preparation, M.K. and A.B. ; writing--review and editing, M.K. and P.A. ; visualization, O.H. ; supervision, M.K. and P.A. ; funding acquisition, P.A. All authors have read and agreed to the published version of the manuscript. This research was supported by the National Research, Development, and Innovation Office of Hungary under the \"Frontline\" Research Excellence Program, (Grant. No. KKP 133827), Project no.