Entropy production in the quantum measurement of continuous observables

The von Neumann entropy produced during the quantum measurement of an observable with a continuous spectrum is calculated and shown to depend on the accuracy ε of the measurement as DIln(1/ε) where DI is the so-called information dimension of the probability distribution of the eigenvalues observed if the system is in a given quantum state. Accordingly, the thermodynamic cost is larger for finer accuracy, unless the spectrum is purely discrete, in which case DI=0.