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4.3: Imperfect Measurements
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_(Kok)/04%3A_Mixed_States_and_the_Density_Operator/4.03%3A_Imperfect_Measurements
While the total number of ideal measurement outcomes, modelled by $P_{m}$ , must be identical to the dimension of the Hilbert space (ignoring the technical complication of degeneracy), this is no lon...While the total number of ideal measurement outcomes, modelled by $P_{m}$ , must be identical to the dimension of the Hilbert space (ignoring the technical complication of degeneracy), this is no longer the case for the POVM described by $E_{m}$ ; there can be more or fewer measurement outcomes, depending on the physical details of the measurement apparatus.

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