on of the Hilbert space for the system of study, and by also using the right operator, or in logical sense, asking the right question. The necessity of measurement to be weak is apparent from the operator used and the freedom left for post-selecting the final state, as weak measurement only perturb the quantum system slightly. The conclusion was there are no two particles in one box, but that does not imply that there is one particle in each box. The latter statement comes from sharp measurement that assigns a definite state for a particle. We can reproduce the quantum pigeonhole effect for energy levels from the using the same argument in Section 2 on the decomposition of (5), and thereby have no two particles have the same energy. This fuzzy energy the ensemble of (5) seems really interesting for quantum logic and computing. Somehow, we let the ensemble carry more information that it would usually do in the case of sharp measurement. The post-selected state for the ensemble shows clearly how we gain a fuzzy notion of occupation number, no constraint on the system violated, energy remains conserved, and the number of particles remains the same. Nevertheless we no longer are able to specify the energy state for each particle, as they are no longer separate and in some form of correlation. This can be applied to any state, not just for energies. We mentioned energies because they are of an interest for statistical mechanics study. As what was done in Section 5 when the ensemble was immersed in a reservoir, or even in an open system. As long as the number of particles and energy levels is limited and no sharp measurement is preformed we still can notice the quantum pigeonhole effect. This effect shows that a particle can carry more than one energy value, therefore the ensemble can be made to carry more information than its standard capacity; when we don’t assign each particle with a specific energy. This could show usefulness in quantum computing. For example, a weak measurement could be a slight shift in energy levels in the post-selected state, by changing the properties of the well enclosing the ensemble increasing the capacity of information processing or storage by creating quantum correlations as seen above. However, this effect could only hold when the system has discrete states, in energy bands forming from very large ensembles, this effect no longer exists, as we have almost continuous energies in the ensemble.
7. Conclusion Remarks
With an ensemble composed of few quantum particles, a new quantum effect can be observed. Provided we only gently measure the system (i.e. avoid assigning each particle a single state; by sharp measurement), quantum interference allows extra configurations (extra ways) to arrange the ensemble. Particles loose separability and we get a different counting logic, giving rise to the quantum pigeonhole principle. This is a new insight to the quantum world, supporting Copenhagen interpretation. This new statistical mechanics could be of an interest in quantum computing and information theory. This effect could only be considered when the ensemble is not very large, as the effect is ultra-quantum, and appears when the energy states are very distinct, not―for example― when energy bands form.
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