I asked a question back in November on Physics.StackExchange, but that didn’t attract any interest from anyone. I started thinking about it again recently and figured out a good solution. The question and answer are explained below.^{a }

**Q**: Is there a good notion of a “diagonal” operator with respect the overcomplete basis of coherent states?

**A**: Yes. The operators that are “coherent-state diagonal” are those that have a smooth Glauber–Sudarshan P transform.

The primary motivation for this question is to get a clean mathematical condition for diagonality (presumably with a notion of “approximately diagonal”) for the density matrix of a system of a continuous degree of freedom being decohered. More generally, one might like to know the intuitive sense in which , , and are all approximately diagonal in the basis of wavepackets, but is not, where is the unitary operator which maps

(1)

(This operator creates a Schrodinger’s cat state by reflecting about .)

For two different coherent states and , we want to require an approximately diagonal operator to satisfy , but we only want to do this if . For , we sensibly expect to be within the eigenspectrum of .

One might consider the negativity of the Wigner-Weyl transform^{b } of the density matrix (i.e. the Wigner phase-space quasi-probability distribution aka the Wigner function) as a sign of quantum coherence, since it is known that coherent superpositions (which are clearly not diagonal in the coherent state basis) have negative oscillations that mark the superposition, and also that these oscillations are destroyed by decoherence.… [continue reading]