Figure 2: Diagram of the general setup.
A decoherence process—also known as a (quantum) channel—can be thought of as an interaction UI of the system A′ with an environment E. In quantum mechanics, the resulting state is the output of the channel 
. In general, B (Bob) may be a smaller or larger system than A′. In the examples below, however, we will focus on the case where A′ and B have the same dimension, corresponding to the case where a fixed system A′=B experiences some interaction with another system E (Eve). The channel’s (in)ability to preserve quantum information—and therefore the amount of decoherence—can be characterized by how well it preserves entanglement between an outside system A and A′. We note that our treatment of theories that go beyond standard quantum mechanics makes no statement whether the environment is an actual physical system, or merely a mathematical Gedanken experiment possibly used to describe an intrinsic decoherence process. In full generality, the experiment consists of a Bell experiment in which a source of decoherence is introduced deliberately. For simplicity, we consider an experiment for the CHSH inequality, although our analysis could easily be extended to any other Bell inequality. In each run, a source prepares the maximally entangled state ΦAA′, where A′ is subsequently exposed to the decoherence process to be tested. We then perform the standard CHSH measurements: system A is measured with probability 1/2 using observables A0=σX and A1=σZ respectively. System B is measured using observables 
and 
with probability 1/2 each. Performing the experiment many times allows an estimate of β=Tr[ρAB(A0⊗B0+A0⊗B1+A1⊗B0−A1⊗B1)].
