# As to the problem of tensor products for quantum logic

```Hi Folks,

I have been researching the notion of tensor products in quantum logic
and found the following for your consideration and comment:```
```
http://plato.stanford.edu/entries/qt-quantlog/
***
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
space
{{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
orthocoherent.
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".

Onward!

Stephen P. King

--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-l...@googlegroups.com.
To unsubscribe from this group, send email to