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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<\infty denote the maximal number of experiments that can be performed to distinguish the states of the system. [632.3.4] Let \{ A_{i}\} _{1}^{m}\subset\mathfrak{A} with i=1,...,m denote the observables in these experiments, and let \eta _{i} (i=1,...,m) be the experimental resolutions or accuracy that can be attained for A_{i}. [632.3.5] Two states \mathsf{z},\mathsf{z}^{{\prime}}\in\mathsf{Z} with

|\left\langle\mathsf{z},A_{i}\right\rangle-\left\langle\mathsf{z}^{{\prime}},A_{i}\right\rangle|=|\left\langle\mathsf{z}-\mathsf{z}^{{\prime}},A_{i}\right\rangle|<\eta _{i}\leq\eta=\max _{{i=1,...,m}}\eta _{i} (24)

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

\mathsf{N}(\mathsf{z};\{ A_{i}\} _{1}^{m};\eta)=\{\mathsf{z}^{{\prime}}\in\mathfrak{A}^{*}:|\left\langle\mathsf{z}-\mathsf{z}^{{\prime}},A_{i}\right\rangle|<\eta _{i},i=1,...,m\} (25)

are \eta-neighborhoods of \mathsf{z} in the weak* topology [17]. [632.3.7] The algebra \mathfrak{M} generated by the elements A_{1},...,A_{m}\in\mathfrak{A} will be called macroscopic algebra.

[632.4.1] In the following 0<\eta _{i}<\infty and 0<\eta=\max _{i}\eta _{i}<\infty will be assumed. [632.4.2] The \eta-neighborhoods of \varepsilon-almost invariant states, i.e. the sets \mathsf{N}(\mathsf{z};\{ A\} _{1}^{m},\eta)\cap\mathsf{B}_{\varepsilon} with \mathsf{z}\in{\mathsf{B}_{0}} for small \varepsilon,\eta\to 0 will be the candidates for local (in time) stationary states.