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We take causal structure to be some relation defined over elements which are understood to be morphisms of some category. An example of such a relation is a domain, another is a directed acyclic graph. Yet another is a string diagram in a symmetric monoidal category.

Panangaden and Martin showed that interval domains are categorically equivalent to (hyperbolic?) spacetimes. This makes the domain a perfect candidated for classical relativistic causality. The planar graphs of the diagrammatical calculus are an enticing candidated for quantum causal structure. Then there is Hardy's causaloid. The classical causal structure seems very realist, as in we might believe in the existence of the set of events for a universe. I have offered the Fischer Impossibility result as a refutation of the naïve existence of this set.

What are the best candidates for a quantum causal structure?

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Why shouldn't the quantum causal structure be identical to the classical causal structure? Are you worried about quantum gravity and information escaping from black holes? Or are you worried about entanglement affecting causality? And why should the Fischer Impossibility result be related to either of these questions? –  Peter Shor Nov 25 '11 at 4:44
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I don't understand what you mean by quantum causal structure. Quantum mechanics is non-signalling so causal structure is the same as in the classical case. –  Joe Fitzsimons Nov 25 '11 at 11:40
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@Joe: The question of causal structure with respect to quantum gravity and black holes is, as far as I am concerned, wide open. However, this question seems to me to have thrown together a paper related to quantum gravity (Hardy's causaloid), a paper on the connection between classical general relativity and a continuous analog of posets (Panangaden and Martin), a mention of the diagrammatic calculus which is too vague to tell exactly what is meant, and what strikes me as a completely unrelated theorem from distributed computing (Fischer). I'm voting to close as not a real question. –  Peter Shor Nov 25 '11 at 17:13
    
@Peter: Sorry Peter, I was referring to the original question, not your comment. I don't dispute that the quantum gravity case is open, but we know so little about that area that it is hardly surprising that it is open. I know some of the results the question refers to, but can't make much sense of what the poster has in mind. I'll cast a virtual vote to close, but since my vote is binding, I won't actually close the question yet. I'd like to give the OP time to actually explain what they mean. If nothing happens in a few days, I'll kill the question. –  Joe Fitzsimons Nov 25 '11 at 17:39
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You can use fIscher impossibility to show that consensus over the set of all processes, ie a notion of a universe, is untenable as it is a direct analogue of group consensus. This is when working under the assumption of a universe of discrete events with a causal structure. We see computer crashes like falling into a black hole. I wrote about that here: cs.mcgill.ca/~bsprot1/EvolvingUniverseFeb24.pdf –  Ben Sprott Nov 25 '11 at 23:50

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up vote 4 down vote accepted

It seems the FLP impossibility result may not hold for quantum systems (see this paper by Helm and section 5 of this paper for a criticism of the first), in which case you don't need any exotic causal structure for quantum mechanics to avoid it.

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in this paper arxiv.org/abs/0903.1133 (section 5) is claimed that Helm results are correct but, (quote) "due to the modification of initialization of the system, and not to quantum mechanics." –  lurscher Nov 26 '11 at 18:08
    
@lurscher: Good catch. I'm not really sure where things stand then. I've updated my answer to include the reference in your comment. –  Joe Fitzsimons Nov 28 '11 at 11:21

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