Sampling Theorems Associated with Singular q-Sturm Liouville Problems

In this paper we introduce three sampling theorems for transformations defined in terms of Jackson q-integration when the kernel of the transformation is a solution or the Green’s function of singular q-Sturm–Liouville problems. We consider the problem when the q-Sturm–Liouville problem is singular either at infinity or at zero with detailed investigations when the singular point is infinity. This approach allows the derivation of sampling representations for transforms whose kernels are linear combinations of q-Bessel functions, not just a single one as previously established.     

Randomized Entangled Mixed States from Phase States

We construct randomized entangled mixed states by using the formalism of phase states for d-dimensional systems (qudits). The randomized entangled mixed states are a special kind of mixed states that exhibit genuine multipartite correlation. Such states are obtained by the application of randomized entangling operators to an arbitrary pair of qudits of a multiqudit system. The study of the entanglement of randomized mixed states is of great importance in quantum computation since any experimental implementation of entangled states in a realistic environment can be made by imperfect entangling gates. We give a brief review of some necessary background about unitary phase operators and phase states of a multi-qudit system. Evolved density matrices arise when qudits of the multi-qudit system interact via a Hamiltonian of Heisenberg type. The randomized entangled states associated with evolved density matrices are derived via the action of an entangling operator on a pair of two qudits {i, j} of the multi-qudit system with some probability p. The randomized entangled mixed states for bipartite, tripartite and multipartite systems are explicitly expressed and their Kraus decomposition properties are discussed.