# Bell-Clauser-Horne-Shimony-Holt Behavior Under Quantum Loss and Decoherence
###### Abstract  
We present a detailed analysis of the effect of quantum loss and decoherence in the Bell-CHSH scenario. Adopting a device-independent approach, we study the change in the bipartite conditional probability distribution, i.e., the behavior of the realized nonlocal box pair when the elements of the entangled qubit pair subjected to independent noisy quantum channels modeled by completely positive maps. As the verification of Bell inequalities is crucial in device-independent quantum cryptography, our considerations are instructive from the perspective of quantum realizations of nonlocal box pairs. We find that the impact of quantum channels cannot be described by an equivalent classical noise channel. device-independent quantum cryptography; Clauser-Horne-Shimony-Holt test; quantum correlations; qubit channels +
Footnote †: journal: _Cryptography_  
0000-0002-122-0000-22000-22000-22000-22000-22000-22000-22000-22000-22000-22000-22000-22000-222000-22000-22000-22000-22000-22000-22000-222000-222000-220  
are those that can be realized using a quantum setup, with an entangled resource and measurements. These form a broader closed convex set. Finally, no-signaling behaviors are even more general; their defining property is that they cannot be used for communication between the parties. They also form a polytope which contains the set of quantum behaviors. The nontrivial facets of the polytope are defined by the no-signaling conditions. Whenever an entanglement-based or device-independent quantum cryptography protocol is realized, it will be subject to imperfections, such as noise, loss, and detector imperfections. Detection loopholes have always been the main issues that arise with practical Bell-inequality-violation testing. Meanwhile, detector inefficiencies can hide nonlocal correlations, turning the behavior into a local one [8; 10]. Loss and decoherence, on the other hand, can decrease the entanglement of the entangled resource, which is required for quantum nonlocality. As also summarized in the review by [PERSON] et al. [1], the tolerance of nonlocal correlations to the addition of various types of noise can be considered an operational measure of nonlocal correlation. This aspect has been extensively studied for different noise types, including white noise [11; 12], local noise [13; 14], and detection inefficiencies [15; 16]. The quantum bit error rate (QBER), which is a key feature of quantum cryptographic protocols, depends on Bell inequality violation. The effect of the atmosphere, detector imperfections, and other relevant factors is extensively studied in the literature on optical satellite quantum cryptography [17; 18; 19], based on particular physical models [20]. From a theoretical point of view, the effect of losses and noise is modeled most generally by quantum channels, i.e., completely positive trace preserving (CPTP) maps. The effect of a local quantum channel on bipartite entanglement has been studied in detail before. Entanglement breaking channels are those CPTP maps which eliminate entanglement when applied to one part of an entangled pair; they have been characterized in a mathematically elegant fashion [21; 22]. In the present work, we approach the question from the Bell inequalities' point of view: we study the effect of the CPTP maps on nonlocal behaviors. The use of CPTP maps is not uncommon in the literature of device-independent quantum cryptography (see, e.g., [23]). The particular effect of CPTP maps on nonclassical correlations, which is the subject of the present contribution, is less broadly discussed in the literature. [PERSON] [24] considers information-theoretic Bell inequalities in a dephasing environment, whereas [PERSON] et al. [25] study the behavior of qubit Bell inequality violations under qubit amplitude damping. These contributions focus on the Bell inequality violations, and not the detailed structural properties of the correlations. [PERSON] et al.