Partial Hasse invariants on splitting models of Hilbert modular varieties

Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-representations attached to Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties $\mathrm{mod}$ $p^m$, extending a previous result of M. Emerton and the authors which required $p$ to be unramified in $F$.Comment: 24 pages, refereed version, index changed from the previous versio

Quantum discord for general two--qubit states: Analytical progress

We present a reliable algorithm to evaluate quantum discord for general two--qubit states, amending and extending an approach recently put forward for the subclass of X--states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of non-classical correlations, namely the geometric discord. We identify the extremally non-classically correlated two--qubit states according to the (normalized) geometric discord, at fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.Comment: 8 pages, 2 figure

Hybridized tetraquarks

We propose a new interpretation of the neutral and charged X,Z exotic hadron resonances. Hybridized-tetraquarks are neither purely compact tetraquark states nor bound or loosely bound molecules but rather a manifestation of the interplay between the two. While meson molecules need a negative or zero binding energy, its counterpart for h -tetraquarks is required to be positive. The formation mechanism of this new class of hadrons is inspired by that of Feshbach metastable states in atomic physics. The recent claim of an exotic resonance in the View the MathML source channel by the D0 Collaboration and the negative result presented subsequently by the LHCb Collaboration are understood in this scheme, together with a considerable portion of available data on X,Z particles. Considerations on a state with the same quantum numbers as the X(5568) are also made