The transition matrix elements for the 0 + → 0 + double beta decays are calculated for 48 Ca, 76 Ge, 82 Se, 100 Mo, 128 T e and 130 T e nuclei, using a δ-interaction. As a guide, to fix the particle-particle interaction strengths, we exploit the fact that the missing symmetries of the mean field approximation are restored in the random phase approximation by the residual interaction. Thus, the T = 1, S = 0 and T = 0, S = 1 coupling strengths have been estimated by invoking the partial restoration of the isospin and Wigner SU (4) symmetries, respectively. When this recipe is strictly applied, the calculation is consistent with the experimental limit for the 2ν lifetime of 48 Ca and it also correctly reproduces the 2ν lifetime of 82 Se. In this way, however, the two-neutrino matrix elements for the remaining nuclei are either underestimated (for 76 Ge and 100 Mo) or overestimated (for 128 T e and 130 T e) approximately by a factor of 3. With a comparatively small variation (< 10%) of the spin-triplet parameter, near the value suggested by the SU(4) symmetry, it is possible to reproduce the measured T 2ν 1/2 in all the cases. The upper limit for the effective neutrino mass, as obtained from the theoretical estimates of 0ν matrix elements, is < m ν > ∼ = 1 eV. The dependence of the nuclear matrix elements on the size of the configuration space has been also analyzed. † Fellow of the CONICET from Argentina * Financial support from CNPq, Brazil is acknowledged

A characterization of multipartite quantum states having N subsystems, based on negativities of matrices obtained by selective partial transposition of state operator, is proposed. The K−way partial transpose with respect to a subsystem is constructed by imposing constraints involving the states of K subsystems of multipartite composite system. The K−way negativity, defined as the negativity of K−way partial transpose, quantifies the K−way coherences of the composite system. For an N-partite system the fraction of K−way negativity (2 ≤ K ≤ N ), contributing to global negativity, is obtained. The entanglement measures for a given state b ρ are identified as the partial K−way negativities of the corresponding canonical state.Positive partial transpose (PPT), first introduced by Peres [1], is the most widely used separability criterion for quantum states. It has been shown to be a necessary and sufficient condition [2] for the separability of qubit-qubit and qubit-qutrit systems. For higher dimensional systems, positive partial transpose is a necessary condition [3,4]. Negativity [5,6] based on Peres Horodecki PPT criterion has been shown to be an entanglement monotone [6,7]. In a multipartite quantum system composed of N subsystems, a single subsystem may be entangled to (N-1)systems in distinctly different ways. For example, in a three qubit system (ABC), the subsystem A can have genuine tripartite entanglement, W-like entanglement, as well as bipartite entanglement with subsystem B or C alone. The negativity of partial transpose of state operator ρ ABC with respect to A, may, thus have distinct contributions that can be related to genuine tripartite or bipartite entanglement. The bipartite entanglement, may in turn be for the pair AB, the pair AC, or both the pairs. In a recent article [8], we have discussed the entanglement of three qubit states using 2−way and 3−way negativities. In this article, a characterization of multipartite quantum states having N subsystems, based on negativities of matrices obtained by selective partial transposition of state operator, is proposed. The K−way partial transpose with respect to a subsystem is constructed by imposing constraints involving the states of K subsystems of multipartite composite system. The K−way negativity (2 ≤ K ≤ N ), defined as the negativity of K−way partial transpose, quantifies the K−way coherences of the composite system. The underlying idea of selective transposition to construct a K−way partial transpose with respect to a subsystem, first presented in ref.[9], shifts the focus from K−subsystems to K−way coherences of the composite system. By K−way coherences, we mean the quantum correlations responsible for GHZ state like entanglement of a K-partite system. For an N-partite entangled state, the negativity of global partial transpose is found to contain contributions from K−way partial transposes (2 ≤ K ≤ N ). Entanglement is invariant under local unitary rotations, whereas, coherences are not so. The elements in the set of states obtained by performin...

A q-deformed analogue of zero-coupled nucleon pair states is constructed and the possibility of accounting for pairing correlations examined. For the single orbit case, the deformed pairs are found to be more strongly bound than the pairs with zero deformation, when a real-valued q parameter is used. It is found that an appropriately scaled deformation parameter reproduces the empirical few nucleon binding energies for nucleons in the 1'/Q orbit and 1gQIq orbit. The deformed pairHamiltonian apparently accounts for many-body correlations, the strength of higher-order force terms being determined by the deformation parameter q. An extension to the multishell case, with deformed zero-coupled pairs distributed over several single particle orbits, has been realized.An analysis of calculated and experimental ground state energies and the energy spectra of three lowermost 0+ states, for even-A Ca isotopes, reveals that the deformation simulates the efFective residual interaction to a large extent.

Partial transposition of state operator is a well known tool to detect quantum correlations between two parts of a composite system. In this letter, the global partial transpose (GPT) is linked to conceptually multipartite underlying structures in a state -the negativity fonts. If K−way negativity fonts with non zero determinants exist, then selective partial transposition of a pure state, involving K of the N qubits (K ≤ N ) yields an operator with negative eigevalues, identifying Kbody correlations in the state. Expansion of GPT interms of K−way partially transposed (KPT) operators reveals the nature of intricate intrinsic correlations in the state. Classification criteria for multipartite entangled states, based on the underlying structure of global partial transpose of canonical state, are proposed. Number of N −partite entanglement types for an N qubit system is found to be 2 N−1 − N + 2, while the number of major entanglement classes is 2 N−1 − 1. Major classes for three and four qubit states are listed. Subclasses are determined by the number and type of negativity fonts in canonical state.

In this letter we propose to quantify three qubit entanglement using global negativity along with K−way negativities, where K = 2 and 3. The principle underlying the definition of K−way negativity for pure and mixed states of N −subsystems is PPT sufficient condition. However, K−way partial transpose with respect to a subsystem is defined so as to shift the focus to K−way coherences instead of K subsystems of the composite system. PACS numbers:Quantum entanglement is not only a fascinating aspect of multipartite quantum systems, but also a physical resource needed for quantum communication, quantum computation and information processing in general. Bipartite entanglement is well understood, however, many aspects of multipartite entanglement are still to be investigated. Peres [1] and the Horedecki [2,3,4] have shown a positive partial transpose (PPT) of a bipartite density operator to be a sufficient criterion for classifying bipartite entanglement. Negativity [5,6] based on Peres Horodecski criterion has been shown to be an entanglement monotone [7,8,9]. Negativity is a useful concept being related to the eigenvalues of partially transposed state operator and can be calculated easily. In this letter, we define 2−way and 3−way negativities and propose a classification of three qubit states based on measures related to global, 2−way and 3−way negativities. General definition of K−way negativities for pure and mixed states of N −subsystems is given in ref [10]. The K−way partial transpose with respect to a subsystem is defined so as to shift the focus to K−way coherences instead of K subsystems of the composite system. While pure K− partite entanglement of a composite system is generated by K−way coherences, K−partite entanglement can, in general, be present due to (K − 1) way coherences as well. I. GLOBAL NEGATIVITYThe Hilbert space, To measure the overall entanglement of a subsystem p, we shall use twice the negativity as defined by Vidal and Werner [6], and call it global Negativity N (2) * Electronic address: shelly@uel.br † Electronic address: nsharma@uel.br

Boson creation operators constructed from linear combinations of q-deformed zero coupled nucleon pair operators acting on the nucleus (A,0), are used to derive pp-RPA equations. The solutions of these equations are the pairing vibrations in (A+2) nuclei.For the 0 + 1 and 0 + 2 states of the nucleus 208 Pb, the variations of relative energies and transfer cross-sections for populating these states via (t,p) reaction, with deformation parameter τ have been analysed. For τ = 0.405 the experimental excitation energy of 4.87MeV and the ratio σ(0 + 2 ) σ(0 + 1 ) = 0.45 are well reproduced. The critical value of pairing interaction strength for which phase transition takes place, is seen to be lower for deformed zero-coupled nucleon pair condensate with τ real, supporting our earlier conclusion that the real deformation simulates the two-body residual interaction. For τ purely imaginary a stronger pairing interaction is required to bring about the phase transition. The effect of imaginary deformation is akin to that of an antipairing type repulsive interaction.Using deformed zero coupled quasi-particle pairs, a deformed version of Quasi-boson approximation for 0 + states in superconducting nuclei is developed. For the test model of 20 particles in two shells, the results of q-deformed boson and quasi-boson approximations have been compared with exact results. It is found that the deformation effectively takes into account the anharmonicities and may be taken as a quantitative measure of the correlations not being accounted for in a certain approximate treatment.

Local unitary invariance and the notion of negativity fonts are used as the principle tools to construct four qubit invariants of degree 8, 12, and 24. A degree 8 polynomial invariant that is non-zero on pure four qubit states with four-body quantum correlations and zero on all other states, is identified. Classification of four qubit states into seven major classes, using criterion based on the nature of correlations, is discussed. * Electronic address: shelly@uel.br † Electronic address: nsharma@uel.br

The effect of the "B" term in the interaction −χQ · Q(1 + B τ (1) · τ (2)), which was previously considered in the 0p shell, is now studied in a larger space which includes ∆N = 2 excitations. We still get a collapse of low-lying states below the conventional J = 0 + ground state when B is made sufficiently negative.