Hi Folks,

        I have been researching the notion of tensor products in quantum logic 
and found the following for your consideration and comment:

7. Composite Systems
Some of the most puzzling features of quantum mechanics arise in connection 
with attempts to describe compound physical systems. It is in this context, for 
instance, that both the measurement problem and the non-locality results 
centered on Bell's theorem arise. It is interesting that coupled systems also 
present a challenge to the quantum-logical programme. I will conclude this 
article with a description of two results that show that the coupling of 
quantum-logical models tends to move us further from the realm of Hilbert space 
quantum mechanics. 
The Foulis-Randall Example
A particularly striking result in this connection is the observation of Foulis 
and Randall [1981] that any reasonable (and reasonably general) tensor product 
of orthoalgebras will fail to preserve ortho-coherence. Let A5 denote the test 
{{a,x,b}, {b,y,c}, {c,z,d}, {d,w,e}, {e,v,s}}
consisting of five three-outcome tests pasted together in a loop. This test 
space is by no means pathological; it is both ortho-coherent and algebraic. 
Moreover, it admits a separating set of dispersion-free states and hence, a 
classical interpretation. Now consider how we might model a compound system 
consisting of two separated sub-systems each modeled by A5. We would need to 
construct a test space B and a mapping ⊗ : X × X → Y = ∪B satisfying, 
minimally, the following;
a.      For all outcomes x, y, z ∈ X, if x⊥y, then x⊗z ⊥ y⊗z and z⊗x ⊥ z⊗y,
b.      For each pair of states α, β ∈ ω(A5), there exists at least one state ω 
on B such that ω(x⊗y) = α(x)β(y), for all outcomes x, y ∈ X.
Foulis and Randall show that no such embedding exists for which B is 
Aerts' Theorem
Another result having a somewhat similar force is that of Aerts [1982]. If L1 
and L2 are two Piron lattices, Aerts constructs in a rather natural way a 
lattice L representing two separated systems, each modeled by one of the given 
lattices. Here “separated” means that each pure state of the larger system L is 
entirely determined by the states of the two component systems L1 and L2. Aerts 
then shows that L is again a Piron lattice iff at least one of the two factors 
L1 and L2 is classical. (This result has recently been strengthened by Ischi 
[2000] in several ways.)
The thrust of these no-go results is that straightforward constructions of 
plausible models for composite systems destroy regularity conditions 
(ortho-coherence in the case of the Foulis-Randall result, orthomodularity and 
the covering law in that of Aerts' result) that have widely been used to 
underwrite reconstructions of the usual quantum-mechanical formalism. This puts 
in doubt whether any of these conditions can be regarded as having the 
universality that the most optimistic version of Mackey's programme asks for. 
Of course, this does not rule out the possibility that these conditions may yet 
be motivated in the case of especially simple physical systems.
        What does this imply? Bruno wrote that " If *you* can demonstrate that 
arithmetical quantum logic have no tensor product enough "coherent" for 
allowing concurrency, then you have refuted DM." but given the results 
discussed here I think that there might be some way to wiggle out of this by 
attacking the notion of "separated".


Stephen P. King

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