Quantum counting and writing: Exotic quantum statistics and the translation of quantum texts

Two big topics are presented in this work: generalized quantum statistics and generalizations of quantum cloning.; The general principles of quantum mechanics do not prohibit the existence of particles obeying generalized (that is, non-Bose and non-Fermi) statistics.; In the first-quantized approach the notions of identicality and indistinguishability for quantum particles, based on the action of the permutation groups, are introduced. The 35-year-old HST classification of the allowed forms of quantum statistics is reviewed.; Closed-form expressions for the Grand Canonical Partition Functions of indistinguishable parabosons and parafermions of any order are presented, and extended to systems that obey the more general statistics in the HST classification.; A new type of fractional statistics called fuzzy statistics is introduced and classified. Examples include the first generalized quantum statistics which smoothly interpolate between bosons and fermions while maintaining extensivity and positive counting.; A characterization is given of second quantized realizations for which the permutation groups act unitarily on the Fock space of states, ghost states are absent and matrix elements obey a clustering condition. The connection with the HST classification is explained.; To better understand the aspects of quantum mechanics responsible for the no-cloning theorem two types of entangled deformations of quantum cloning, called enscription and translation are introduced and investigated. The effectiveness of enscription as a resource for "probabilistic cloning" is emphasized and a complete characterization of all translatable texts is presented.