Are objective measurement values compatible with unitarity?

For an actual measurement to be reliable, it suffices that the measurement value reproduces the objective expectation value with high probability, within the precision of the measurement. Thanks to the law of large numbers, this remains the case with unitary dynamics.

The quantum mechanical uncertainties in the reading of a macroscopic device are tiny. The measurement device is in local equilibrium, thus the pointer has a density matrix corresponding to a grand canonical ensemble. One can calculate the order of magnitude of the fluctuations according to the rules of statistical mechanics, and verify that they play no significant role.

The relative uncertainty is proportional to N-1/2, where N is the number of particles in the pointer. For a good pointer N will not be much less than 1020.

This is indeed the whole secret of statistical mechanics. Without the law of large numbers thermodynamics would be impossible…

This is also the reason why the Copenhagen interpretation has to insist on a classical measurement device - a small quantum system is too unreliable to read.

Arnold Neumaier (
A theoretical physics FAQ