Erasing one bit of information has a minimum thermodynamic cost of E = k_B T ln(2) under standard conditions, a fundamental limit proposed by Rolf Landauer in 1961 and experimentally verified in 2012.
I was reminded of this neat little fact while writing the background chapter of my thesis.
If you let the initial position/momentum of the harmonic oscillator be random, x(t) and p(t) define a stochastic process. Cov(x,p) = 0 but they are not independent because p = m dx/dt!