Dealing with environment in a CHSH game I am reading arxiv:1209.0448. I understand that my questions
could be highly trivial. I would appreciate if anyone helps me to resolve
my confusions.
In a CHSH game, Alice and Bob cannot have classical communication channel between them. According to the case of the second paragraph of page 6, Eve chooses to stop playing with either Alice or Bob selectively without notifying the other party. Let us consider the case when Alice has stopped while Bob is still playing.
I am confused whether 'not having a classical communication channel with Alice' also includes 'not having access to the environment Alice is in'. Because, if Alice receives the input it will change the entropy of Alice's system which eventually effects the entropy of Alice's environment. If Bob cannot communicate with Alice but has access to the environment where Alice is in, he may try to analyze the thermodynamic footprint and guess what is going on. I understand that ideally Alice and Bob can be in two isolated environments but in real life it is not feasible, right?
 A: Not having a classical communication channel with Alice means as you say that they are isolated and restricted to performing local operations on their systems (located in spatially separated labs if you wish). 
As long as the local operations are unitary and no systems are traced out (no noisy evolution on Alice's site), the "environment" should be completely ignorant of Alice's operations. And even if she does, in any case those operations commute with Bob's system which means that nothing he can do locally could help him guess what's happening at Alice's site.
Of course it's just a model, and if you want to implement it and guarantee that all things are behaving as predicted you'll have a hard time. In fact, no loophole-free CHSH violation has been verifed as of today.
I'm not sure if it addresses your question, but maybe it helps you a little.
A: Consider the following:  CHSH was derived (By CHSH) as an extension of Bell's original inequalities.  However CHSH partitioned their classical space into two regions
B+C and B-C  (I will let you put in LHV and filter angles).  Now this means the two vectors are orthogonal if B and C are orthogonal, and the CHSH derivation does not take into account Heisenberg Uncertainty (Classical equation).
If a spin has two axes of quantization, then you can prove it is impossible, because of Heisenberg, to measure both B+C and B-C simultaneously.  Therefore Nature will not violate CHSH  (It does violate Bell's original inequalities)
Now I am certain this is cryptic, but I have talked about it on my blog and recently have simulated the complete quantum correlation -cos(theta) with a local realistic model.
http://quantummechanics.mchmultimedia.com/
