Saturday, April 25, 2015

Don't confuse necessary and sufficient conditions

In Carl Caves recent UQ Quantum Science Seminar "Quantum metrology meets Quantum Information Science" as an aside he also made an important side point.

People often erroneously assume that the converse of a statement is true (i.e.  A implies B means that B implies A). This came up because a few referees had said that some of the results he presented in the seminar were "obvious". Roughly speaking, this concerns the issue of trying to determine what input quantum state to an interferometer will produce a "physical" output state. He found that the input state had to be "physical" (by some well-defined technical criteria). Showing this is non-trivial. However, it is obvious a physical input state is sufficient to produce a physical output state. But, that does not mean it is necessary. Showing this turned out to be quite non-trivial.

I can immediately think of two other cases where scientists made similar errors of conflating necessary and sufficient conditions.

The first case is the existence of quasi-crystals. It was well known that a periodic array of atoms is sufficient to produce sharp diffraction peaks. However, many people erroneously assumed that this was also necessary. I emphasise this point when I teach undergraduates about quasi-crystals.

The second concerns Angle-dependent MagnetoResistance Oscillations in quasi-two-dimensional metals. Not long after their experimental discovery AMRO was explained in terms of a coherent interlayer transport and a three-dimensional Fermi surface. It was subsequently more or less assumed that observing AMRO was evidence for a three-dimensional Fermi surface.  However, in 1998, Perez Moses and I showed that a 3D Fermi surface was not necessary for the existence of AMRO.

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