In weak measurement thought experiment, an ensemble consists of M quantum particles and N states. We observe that separability of the particles is lost, and hence we have fuzzy occupation numbers for the particles in the ensemble. Without sharply measuring each particle state, quantum interferences add extra possible configurations of the ensemble, this explains the Quantum Pigeonhole Principle. This principle adds more entropy to the system; hence the particles seem to have a new kind of correlations emergent from particles not having a single, well-defined state. We formulated the Quantum Pigeonhole Principle in the language of abstract Hilbert spaces, then generalized it to systems consisting of mixed states. This insight into the fundamentals of quantum statistical mechanics could help us understand the interpretation of quantum mechanics more deeply, and possibly have implication on quantum computing and information theory.
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