Parity measurements, decoherence and spiky Wigner functions

Notwithstanding radical conceptual differences between classical and quantum mechanics, it is usually assumed that physical measurements concern observables common to both theories. Not so with the eigenvalues (+/-1) of the parity operator. The effect of such a measurement on a mixture of even and odd states of the harmonic oscillator is akin to separating at a single stroke a pair of shuffled card decks: the result is a set of definite parity, though otherwise mixed. Here we derive the general form of a parity collapsed state, whether pure or mixed. The signature of positive or negative parity is a corresponding spike in the Wigner function which is sharpened by decoherence. We conjecture that states with pure parity always have negative values in their Wigner functions.