“On the Uncertainty of the Ordering of Nonlocal Wavefunction
Collapse when Relativity is Considered,”

by Chris D. Richardson and Jonathan P. Dowling

This preprint
on the ArXiv by my former PhD student, Chris Richardson,
and me, is not getting enough of the well-deserved publicity that it warrants
so here I am to shamelessly promote it. (That and Chris will shortly be looking
for a job.) This is really 90% Chris’s work with 10% motivational pep talks
from me to him. In a previous blog post, “On
the Curious Consistency of Non-Relativistic Quantum Theory with Non-Quantum
Relativity Theory,” I blathered on about how odd it was that
non-relativistic quantum theory always seemed to be consistent with ordinary
relativity theory; even though we have no right to expect that these two theories
should be consistent and every right to believe they should flat out contradict
each other. This curiousness Nicolas Gisin calls the “tension” between the two
theories and Gisin has even done an
experiment of the EPR
type with a well separated Alice and Bob, but with Bob placed in a moving
reference frame (compared to Alice) to try to measure the ‘speed of collapse’
of the two-particle wave function. Gisin and his group conclude that the speed
of collapse is some 10,000 times faster than the speed of light (which is
consistent with infinitely fast).

This experiment motivated Chris and I to think about a
closely related problem; a problem that in fact motivated Gisin’s experiment in
the first place. In non-relativistic quantum theory, in an EPR experiment, if
Alice makes a measurement on her particle then the state of Bob’s particle is
supposed to collapse ‘instantaneously and simultaneously’ to the result
anti-correlated to Alice’s measurement (if they share, say, a spin-singlet
state). But words like ‘instantaneous’ and — Heaven forbid! — ‘simultaneous’
are heresy in non-quantum relativity theory.

This thought experiment gives rise to a purported paradox.
If in one reference frame Alice measures first and collapses Bob’s state there
can always be an observer in a different inertial frame who thinks Bob measured
first and collapsed Alice’s state. The paradox then may be stated, “Who

*really*collapsed*whom*first?” This curious swapping of temporal order is the paradox.
Now if I was David Deutsch I would tell you to avoid all
this collapse nonsense and to instead embrace the Many Worlds Interpretation of
Quantum Mechanics but instead Chris and I decided to push this paradox into a
small logical corner where we could beat the heck out of it with a purely Copenhagen Gedanken
experimental analysis.

The conclusion of our short paper, now in referee limbo in
some journal I’d rather not mention (so as to avoid the once and future lawsuits)
is that

The paroxysm of paradoxism is swept squarely under a round rug.

*Nature*—*not*the journal! — deploys a type of quantum-mechanical cloaking device upon the experiment to keep the paradox from arising in the first place.The paroxysm of paradoxism is swept squarely under a round rug.

The crux of our argument is that Alice and Bob’s
measurements cannot be made with infinite precision but are constrained by the
Heisenberg uncertainty principle — particularly the notorious energy-time
uncertainty principle. Since energy and time are not relativistically invariant
quantities, different observers in different reference frames must transform
their uncertainty principles accordingly.

Therein lies the rug.

To quote the conclusion of our paper; “The uncertainty in
time always outruns the time difference induced by the change in reference
frames. Neither Alice nor Bob will ever, with certainty, observe the two
measurements swap temporal order.”

Paradox, schmÃ¤radox!

The curious consistency of quantum theory and relativity
theory hides again….