A novel approach to entanglement, based on the Gelfand-Naimark-Segal (GNS) construction, is introduced. It considers states as well as algebras of observables on an equal footing. The conventional approach to the emergence of mixed from pure ones based on taking partial traces is replaced by the more general notion of the restriction of a state to a subalgebra. For bipartite systems of nonidentical particles, this approach reproduces the standard results. But it also very naturally overcomes the limitations of the usual treatment of systems of identical particles. This GNS approach seems very general and can be applied for example to systems obeying para-and braid-statistics including anyons.

We obtain a formula for the determinant of a block Toeplitz matrix associated with a quadratic fermionic chain with complex coupling. Such couplings break reflection symmetry and/or charge conjugation symmetry. We then apply this formula to compute the Rényi entropy of a partial observation to a subsystem consisting of contiguous sites in the limit of large size. (2008)]. A striking feature of our formula for the entanglement entropy is the appearance of a term scaling with the logarithm of the size. This logarithmic behavior originates from certain discontinuities in the symbol of the block Toeplitz matrix. Equipped with this formula we analyze the entanglement entropy of a Dzyaloshinski-Moriya spin chain and a Kitaev fermionic chain with long-range pairing.

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 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 the study of time evolution of subsystems emerging from restriction to subalgebras. Also, problems related to anomalies and quantum epistemology are discussed. *

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In this paper we complete the study on the asymptotic behaviour of the entanglement entropy for Kitaev chains with long range pairing. We discover that when the couplings decay with the distance with a critical exponent new properties for the asymptotic growth of the entropy appear. The coefficient of the leading term is not universal any more and the connection with conformal field theories is lost.We perform a numerical and analytical approach to the problem showing a perfect agreement. In order to carry out the analytical study, a new technique for computing the asymptotic behaviour of block Toeplitz determinants with discontinuous symbols has been developed.

Gribov's observation that global gauge fixing is impossible has led to suggestions that there may be a deep connection between gauge-fixing and confinement. We find an unexpected relation between the topological non-triviality of the gauge bundle and coloured states in SU (N ) Yang-Mills theory, and show that such states are necessarily impure. We approximate QCD by a rectangular matrix model that captures the essential topological features of the gauge bundle, and demonstrate the impure nature of coloured states explicitly. Our matrix model also allows the inclusion of the QCD θ-term, as well as to perform explicit computations of low-lying glueball masses. This mass spectrum is gapped.Since an impure state cannot evolve to a pure one by a unitary transformation, our result shows that the solution to the confinement problem in pure QCD is fundamentally quantum informationtheoretic.

It is by now clear that infrared sector of QED has an intriguingly complex structure. Based on earlier pioneering works on this subject, two of us recently proposed a simple modification of QED by constructing a generalization of the U (1) charge group of QED to the "Sky" group incorporating the known spontaneous Lorentz violation due to infrared photons, but still compatible in particular with locality [1]. There it was shown that the "Sky" group is generated by the algebra of angle dependent charges and a study of its superselection sectors has revealed a manifest description of spontaneous breaking of Lorentz symmetry. We further elaborate this approach here and investigate in some detail the properties of charged particles dressed by the infrared photons. We find that Lorentz violation due to soft photons may be manifestly codified in an angle dependent fermion mass modifying therefore the fermion dispersion relations. The fact that the masses of the charged particles are not Lorentz invariant affects their spin content too.Time dilation formulae for decays should also get corrections. We speculate that these effects could be measured possibly in muon decay experiments. 1

There are several instances where quantum anomalies of continuous and discrete classical symmetries play an important role in fundamental physics. Examples come from chiral anomalies in the Standard Model of fundamental interactions and gravitational anomalies in string theories. Their generic origin is the fact that classical symmetries may not preserve the domains of quantum operators like the Hamiltonian. In this work, we show by simple examples that anomalous symmetries can often be implemented at the expense of working with mixed states having non-zero entropies. In particular there is the result on color breaking by non-abelian magnetic monopoles. This anomaly can be rectified by using impure states. We also argue that non-abelian groups of twisted bundles are always anomalous for pure states sharpening an earlier argument of Sorkin and Balachandran [3]. This is the case of mapping class groups * bal@phy.syr.edu † amilcarq@unb.br 1 of geons [3] indicating that large diffeos are anomalous for pure states in the presence of geons. Nevertheless diffeo invariance may be restored by using impure states. This work concludes with examples of these ideas drawn from molecular physics.The above approach using impure states is entirely equivalent to restricting all states to the algebra of observables invariant under the anomalous symmetries. For anomalous gauge groups such as color, this would mean that we work with observables singlet under global gauge transformations. For color, this will mean that we work with color singlets, a reasonable constraint.