In [1], some of the statements concerning odd paragrassmann algebras are misleading, due to the presence of negative values of the deformations of integers [n] q . We hereby redefine the odd algebras not only to ensure that all deformations [n] q are positive, but also to provide a unified treatment of both even and odd algebras, as can be done in the representation theory of the q-oscillator, where q is a primitive root of unity [2].Changes in section 3 'The essential paragrassmann algebra'. The leading paragraphs must be changed to the following.'In this paper, we consider as an observation set X the paragrassmann algebra k for k even [3][4][5][6]. We recall that a Grassmann (or exterior) algebra of a given vector space V over a field is the algebra generated by the exterior (or wedge) product for which all elements are nilpotent, θ 2 = 0. Paragrassmann algebras are generalizations for which, given an even number k > 2, all elements obey θ k = 0, where k = k/2 for even k. For a given k, we define k as the linear span of {1, . . . , θ n , . . . , θ k −1 } and of their respective conjugates θ n : here θ is a paragrassmann variable satisfying θ k = 0. Variables θ and θ do not commute:

We propose a quantum key distribution protocol using Greenberger Horne Zeilinger tripartite coherent states. The sender and the receiver share similar key by exchanging the correlation coherent states, without basis reconciliation. This allows the protocol to have a transmission efficiency of 100% in a perfect quantum channel. The security of the protocol is ensured by tripartite coherent states correlation and homodyne detection, which allows to detect any eavesdropping easily.

Coherent states for power-law potentials are constructed using generalized Heisenberg algabras. Klauder's minimal set of conditions required to obtain coherent states are satisfied.The statistical properties of these states are investigated through the evaluation of the Mandel's parameter. It is shown that these coherent states are useful for describing the states of real and ideal lasers.

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Using the linear entropy as a measure of entanglement, we investigate the effect of a beam splitter on the Perelomov coherent states for the q-deformed Uq(su(2)) algebra. We distinguish two cases: in the classical q → 1 limit, we find that the states become Glauber coherent states as the spin tends to infinity; whereas for q ≠ 1, the states, contrary to the earlier case, become entangled as they pass through a beam splitter. The entanglement strongly depends on the q-deformation parameter and the amplitude Z of the state.