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 ($\pm 1$) 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. The Wigner function should be a sensitive probe for this phenomenon, for it can be interpreted as the expectation value of the parity operator. We here derive the general form of Wigner functions $W_{\pm}$, resulting from an ideal parity measurement on $W(\x)$. Even if $W(\x)$ resembles a classical distribution, $W_{\pm}$ displays a quantum spike, which is positive for $W_+$ and negative for $W_-$. However we conjecture that $W_+$ always has negative values.