So, if two descriptions are equivalent then they have the same referent. If absolutely everything you can say about Aunt Edna is also true of Mrs Johnson, then Aunt Edna is Mrs Johnson. The same holds true of states: if two states are utterly indistinguishable in every way, then they are the same state. This is the reason for the Schroedinger's Cat paradox. Until we open the box the states of the universe in which the cat is dead, and in which it is alive, are equivalent in every respect: thus the cat (from our perspective) is neither alive nor dead. Once we open the box we collapse the wave, and it settles into one state. No cats were hurt in the construction of this metaphor
I think that in the film (more than in the book) the killer, Anton Chigurh, is in a sort of information-box locked off from the rest of the Universe. This means he exists as a wave function, not as a collapsed entity. He neither exists nor unexists. As The Sheriff says 'He is a kind of ghost'. Any observer who might collapse the wave function - by transmitting the information that he exists - is doomed to be killed. Even those who hear about him second hand - who hear someone say 'I have seen him, he exists' - will soon die. And all these killings might have been carried out by other people, specifically a group of anonymous Mexicans, so the dead bodies aren't proof of his existence.
I think the Sheriff understands this at some level, and turns away from looking on Anton, even though this means letting him go free. In another way this makes him into his twin. Almost the last words spoken in the film are when the Sheriff's wife calls him 'Ed Tom'. It sounds as if she called him 'Anton'.
There are exceptions to the 'see and die' rule, but I think they (in a non-scientific way) emphasise the pattern. For instance, Anton interrogates the proprietor of the trailer park where his prey lives. She refuses to tell him where Moss works. I assumed he was about to torture and kill her, but then she says:
'I don't give out information'
And at this he turns away without another words and leaves her unharmed. She doesn't give out information, so she doesn't collapse the wave function.
In another case he flips a coin to randomise whether a man lives or dies. Could the parallel to Schroedinger's cat be any more explicit? Interestingly, I have read two different comments on this scene - one person thought he lived, one thought he died. The wave is not collapsed.
ETA - I also draw your attention to Tommy Lee Jones' other role as Two Face in the Batman films ('Two-face relies on a coin with a scarred head to make all decisions. Every victim, every decision is based on the flip of that coin.')
And finally, at several points key players stop random strangers in the street and buy their silence (along with, funnily enough, their clothes). There is some circular reference in the film and book between collapsing the wave function, 'accounting', money, and the idea that 'you will pay for this'. After having been called 'Ed Tom/Anton' by his wife, the Sheriff says 'I dreamed my father met me on the street and gave me money'.
As soon as the random strangers get the money they start to fight over it, so there is also some parallel with eating the forbidden fruit I think. And some idea that there is a poison to do with money and knowledge, which is spreading through society.