Careful said:
I do not know what MWI is
MWI = Many Worlds Interpretation, a fancy word for assuming that the observer is just as well suffering quantum evolution as everything else, so that an observation does not give rise to a projection, but that everything (including observation) is simply one big unitary evolution.
initial state:
A0, B0 and C0 are the states of the observers before they got involved in the measurement (before they underwent an evolution that entangled them with our system).
|A0)|B0)|C0) (u |a+) + v |a-) )
A "measures":
|B0)|C0) (u |A+)|a+) + v |A-) |a-) )
A+ is the body state of observer A where he saw a + result, and A- is the body state of the observer A where he saw a - result.
B "measures":
B+ is the state of the body of observer B when he's informed about the entire result of the B-measurement (so here we see - see further - than in order to be so informed, B actually has to have the entire B region in his past lightcone, but we're now pretending that this must not be the case).
|C0) u |A+) (|B+) (b+|a+) |b+) + |B-) (b-|a+) |b-))
+ |C0) v |A-)(|B+) (b+|a-) |b+) + |B-) (b-|a-) |b-))
C "measures":
u |A+) (|B+) (b+|a+) (|C+) (a+|b+)|a+) + |C-) (a-|b+)|a-))
+ |B-) (b-|a+) (|C+)(a+|b-)|a+) + |C-)(a-|b-)|a-)) )
+ v |A-)(|B+) (b+|a-) (|C+) (a+|b+)|a+) + |C-) (a-|b+)|a-))
+ |B-) (b-|a-) (|C+)(a+|b-)|a+) + |C-)(a-|b-)|a-) ) )
= |C+) {u |A+) (|B+) (b+|a+) (a+|b+)
+ |B-) (b-|a+) (a+|b-) )
+ v |A-) (|B+) (b+|a-) (a+|b+)
+ |B-) (b-|a-) (a+|b-) )}|a+)
+ |C-) {u |A+) (|B+) (b+|a+) (a-|b+)
+ |B-) (b-|a+) (a-|b-) )
+ v |A-)(|B+) (b+|a-) (a-|b+)
+ |B-) (b-|a-) (a-|b-) )} |a-)
The probability to get C+ is then the total length of the state vector which has the C+ body state as a factor:
|u|^2 (|U11|^4 + |U12|^4) + |v|^2 (U11.U21.U11*.U21*+U22.U12*.U22*.U12)
which is the same result as our first calculation.
Note that a priori we're in the same deep s**t, because if we don't let
A measure, then |A0) stays factored out, and the |A+) and |A-) terms are not
orthogonal anymore, but just add as amplitudes:
B "measures":
|C0) u |A0) (|B+) (b+|a+) |b+) + |B-) (b-|a+) |b-))
+ |C0) v |A0)(|B+) (b+|a-) |b+) + |B-) (b-|a-) |b-))
C "measures":
|C+) |A0) {u (|B+) (b+|a+) (a+|b+)
+ |B-) (b-|a+) (a+|b-) )
+ v (|B+) (b+|a-) (a+|b+)
+ |B-) (b-|a-) (a+|b-) )}|a+)
+ |C-) |A0) {u (|B+) (b+|a+) (a-|b+)
+ (b-|a+) (a-|b-) )
+ v |A-)(|B+) (b+|a-) (a-|b+)
+ |B-) (b-|a-) (a-|b-) )} |a-)
which will us probably give the same result as using projection.
But now we understand why ! The so-called B measurement cannot have taken place completely when C measures, so the B interaction (unitary) has to be split in 2 parts:
the one in the future lightcone of A (BL), and the one in the past lightcone of C (BR). Both interactions (unitary evolutions) BL and BR commute, and BL commutes with C, while BR commutes with A. BL does not commute with A and BR does not commute with C however.
Clearly, my 2-state example is not sufficient in this case to implement these operators, so I give up here for the moment, but I think that this will solve the issue.
In a way, you can say that (typical of the MWI approach) this splitting in BL and BR is part of what you require "detailling the detection procedure".
As far as I can tell, because the only evolution that could possibly influence C (as unitary evolution, using Green's functions all the way within the detector, brain, whatever), is BR, and whatever happens to BL and A should normally factor out, hence not influencing the entanglement of C with the state.
But I should work it out, and I think it's going to take more work and time than I have.
Nevertheless, interesting problem !
