As an exercise I've been trying to pinpoint exactly what is wrong with Plaga's paper. For anyone who doubts that it *is* wrong, note that it proposed 10 years ago an experiment which he said was feasible with what was then state-of-the-art equipment. This technology has now massively advanced. The experiment would guarantee a Nobel prize for anyone who performed it successfully. In that time the paper has been cited in the published literature only 3 times, and never by an experimental physicist. And this is not because the paper was unnoticed by the community at the time, e.g. it was publicised by John Baez, whose writings are widely read.

On careful reading, the paper is just littered with confusions and errors. I guess this explains why no-one bothered to publish a rebuttal; this falls in the class of "not even wrong". Probably the root problem is a confusion about the true nature of decoherence. Decoherence is often presented using the maths of density matrices, so I better explain this briefly:

Density matrices allow you to handle the case when you don't know the exact quantum state. The procedure is to divide your description into a measurable "system" and a complex, not-measurable-in-detail "environment". One can then define the density matrix of the combined system, and "trace out" the uncertain state of the environment, giving a density matrix for the system alone in the absence of information about the environment. A test to see if the system has been decohered by its interaction with the environment is that the off-diagonal terms in the system-only density matrix go to zero. Plaga clearly accepts the usual position that irreversible branching in MWI occurs when decoherence is (FAPP) total.

If you follow this through in Plaga's example, you do indeed find that the density matrix for the states of his trapped ion, |A1> and |A2>, is diagonal, confirming the obvious that once a macroscopic measurement has taken place, we have total decoherence. But what Plaga does in his Eq 8 is to reverse the roles of system and environment (he actually does the algebra wrong but the numerical answer is unaffected). Because at this stage the ion knows nothing about the rest of the lab, he gets a density matrix *for the lab* with large off-diagonal terms, corresponding to a "pure" state:

 (|W1> + |W2>) / sqrt(2).

So far, so correct (after all, in MWI the state is *always* pure).

But he now concludes that decoherence has not yet occurred. *WRONG*. The condition "off-diagonal terms go to zero" is just a sufficient condition for decoherence. It is only necessary if the "system" itself is so simple that it could not decohere without the help of the environment. But Plaga is treating the complex, macroscopic lab as the "system" and that certainly can decohere without the help one more ion. The more basic definition is that decoherence has occured once the states are permanently orthogonal, so you cannot demonstrate quantum interference. Plaga correctly states that |W1> and |W2> *are* permanently orthogonal, but does not realise that this means that decoherence *is* complete, contrary to what he says. Another way to put this is that the observer "Silvia" doesn't need the density matrix in Eq. (8) because she knows for sure already whether she detected the original photon or not, hence whether she is in branch |W1> or |W2>.

Given this, the rest of Plaga's argument is just irrelevant. But he should have noticed that his process blatantly violates the linearity of time evolution, which is one of the fundamental assumptions of MWI QM. This is manifest in his Eq. 6 which associates an excited ion with the |P2> term in which no excitation took place (if you start with a photon in state |P2>, when the photon is guaranteed not to be detected, the ion is never excited). Hence Eq 6 is not a linear superposition of the two possible histories. Hence, if we saw what he predicted, we would actually *disprove* MWI QM, not confirm it as he thinks.

Paddy Leahy

Dr J. P. Leahy, University of Manchester,
Jodrell Bank Observatory, School of Physics & Astronomy,
Macclesfield, Cheshire SK11 9DL, UK
Tel - +44 1477 572636, Fax - +44 1477 571618

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