4 Indistinguishability of states

[632.3.1] Experimental uncertainties limit also the ability to distinguish different states. [632.3.2] Two states are experimentally indistinguishable (or metrologically equivalent) if they cannot be distinguished by measurements. [632.3.3] Let m<∞ denote the maximal number of experiments that can be performed to distinguish the states of the system. [632.3.4] Let Ai1m⊂A with i=1,…,m denote the observables in these experiments, and let ηi (i=1,…,m) be the experimental resolutions or accuracy that can be attained for Ai. [632.3.5] Two states z,z′∈Z with

for all i=1,…,m are called metrologically equivalent or experimentally indistinguishable with respect to the observables A1,…,Am. [632.3.6] The sets of indistinguishable states

are η-neighborhoods of z in the weak* topology [17]. [632.3.7] The algebra M generated by the elements A1,…,Am∈A will be called macroscopic algebra.

[632.4.1] In the following 0<ηi<∞ and 0<η=maxi⁡ηi<∞ will be assumed. [632.4.2] The η-neighborhoods of ε-almost invariant states, i.e. the sets N⁢z;A1m,η∩Bε with z∈B0 for small ε,η→0 will be the candidates for local (in time) stationary states.