We consider the issue of statistics for identical particles or fields in $\kappa$-deformed spaces, where the system admits a symmetry group $G$. We obtain the twisted flip operator compatible with the action of the symmetry group, which is relevant for describing particle statistics in the presence of the noncommutativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the ordering prescription.

We elaborate on the role of quantum statistics in twisted Poincaré invariant theories. It is shown that, in order to have twisted Poincaré group as the symmetry of a quantum theory, statistics must be twisted. It is also confirmed that the removal of UV-IR mixing (in the absence of gauge fields) in such theories is a natural consequence.

It has been known for some years that entanglement entropy obtained from partial trace does not provide the correct entanglement measure when applied to systems of identical particles. Several criteria have been proposed that have the drawback of being different according to whether one is dealing with fermions, bosons, or distinguishable particles. In this Letter, we give a precise and mathematically natural answer to this problem. Our approach is based on the use of the more general idea of the restriction of states to subalgebras. It leads to a novel approach to entanglement, which is suitable to be used in general quantum systems and especially in systems of identical particles. This settles some recent controversy regarding entanglement for identical particles. The prospects for applications of our criteria are wide ranging, from spin chains in condensed matter to entropy of black holes.

We present a general approach to quantum entanglement and entropy that is based on algebras of observables and states thereon. In contrast to more standard treatments, Hilbert space is an emergent concept, appearing as a representation space of the observable algebra, once a state is chosen. In this approach, which is based on the Gel'fand-Naimark-Segal construction, the study of subsystems becomes particularly clear. We explicitly show how the problems associated with the partial trace for the study of entanglement of identical particles are readily overcome. In particular, a suitable entanglement measure is proposed that can be applied to systems of particles obeying Fermi, Bose, para-, and even braid-group statistics. The generality of the method is also illustrated by study of the time evolution of subsystems emerging from restriction to subalgebras. Problems related to anomalies and quantum epistemology are also discussed.