Crystalline Hodge cycles and Shimura curves in positive characteristics

In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization of curves in positive characteristics which are reduction of Shimura curve over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian fourfolds over k, char k=p. We show that some conditions on the crystalline Hodge cycles over such a curve imply that this curve can be lifted to a Shimura curve.Comment: 21 pages. Comments welcom

l-adic Monodromy and Shimura curves in positive characteristics

In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization, in terms of geometry mod p, of curves in positive characteristics which are reduction of Shimura curves over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian varieties over k, char k=p. We show that some conditions on the l-adic monodromy over such a curve imply that this curve can be lifted to a Shimura curve.Comment: 8 pages. Comments are welcome

Tensor decomposition of isocrystals characterizes Mumford curves

We seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics via characterizing curves in positive characteristics which are reduction of Shimura curve over $\mathbb{C}$. In this paper, we study the liftablity of a curve in the moduli space of principally polarized abelian varieties over $k, \text{char} k=p$. We show that in the generic ordinary case, some tensor decomposition of the isocrystal associated to the family imply that this curve can be lifted to a Shimura curve.Comment: 17 pages. Comments are welcom

On Rigidity of hypersurfaces with constant curvature functions in warped product manifolds

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work \cite{Mon, Brendle,BE} considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss-Bonnet curvatures $L_k$. To achieve this, we develop some new kind of Newton-Maclaurin type inequalities on $L_k$ which may have independent interest.Comment: 24 pages, Ann. Glob. Anal. Geom. to appea

Annealed asymptotics for Brownian motion of renormalized potential in mobile random medium

Motivated by the study of the directed polymer model with mobile Poissonian traps or catalysts and the stochastic parabolic Anderson model with time dependent potential, we investigate the asymptotic behavior of $$\mathbb{E}\otimes\mathbb{E}_0\exp\left\{\pm\theta\int^t_0\bar{V}(s,B_s)ds\right\}\qquad (t\to\infty)$$ where $\th>0$ is a constant, $\overline{V}$ is the renormalized Poisson potential of the form $$\overline{V}(s,x)=\int_{\mathbb{R}^d}\frac{1}{|y-x|^p}\left(\omega_s(dy)-dy\right),$$ and $\omega_s$ is the measure-valued process consisting of independent Brownian particles whose initial positions form a Poisson random measure on $\mathbb{R}^d$ with Lebesgue measure as its intensity. Different scaling limits are obtained according to the parameter $p$ and dimension $d$. For the logarithm of the negative exponential moment, the range of $\frac{d}{2}<p<d$ is divided into 5 regions with various scaling rates of the orders $t^{d/p}$, $t^{3/2}$, $t^{(4-d-2p)/2}$, $t\log t$ and $t$, respectively. For the positive exponential moment, the limiting behavior is studied according to the parameters $p$ and $d$ in three regions. In the sub-critical region ($p<2$), the double logarithm of the exponential moment has a rate of $t$. In the critical region ($p=2$), it has different behavior over two parts decided according to the comparison of $\theta$ with the best constant in the Hardy inequality. In the super-critical region $(p>2)$, the exponential moments become infinite for all $t>0$.Comment: 43 page

Quantum speed limits for Bell-diagonal states

Bounds of the minimum evolution time between two distinguishable states of a system can help to assess the maximal speed of quantum computers and communication channels. We study the quantum speed limit time of a composite quantum states in the presence of nondissipative decoherence. For the initial states with maximally mixed marginals, we obtain the exactly expressions of quantum speed limit time which mainly depend on the parameters of the initial states and the decoherence channels. Furthermore, by calculating quantum speed limit time for the time-dependent states started from a class of initial states, we discover that the quantum speed limit time gradually decreases in time, and the decay rate of the quantum speed limit time would show a sudden change at a certain critical time. Interestingly, at the same critical time, the composite system dynamics would exhibit a sudden transition from classical to quantum decoherence.Comment: 5 pages, 2 figure

Enhancing entanglement trapping by weak measurement and quantum measurement reversal

In this paper, we propose a scheme to enhance trapping of entanglement of two qubits in the environment of a photonic band gap material. Our entanglement trapping promotion scheme makes use of combined weak measurements and quantum measurement reversals. The optimal promotion of entanglement trapping can be acquired with a reasonable finite success probability by adjusting measurement strengths.Comment: welcome to commen

One-step implementation of the Fredkin gate via quantum Zeno dynamics

We study one-step implementation of the Fredkin gate in a bi-modal cavity under both resonant and large detuning conditions based on quantum Zeno dynamics, which reduces the complexity of experiment operations. The influence of cavity decay and atomic spontaneous emission is discussed by numerical calculation. The results demonstrate that the fidelity and the success probability are robust against cavity decay in both models and they are also insensitive to atomic spontaneous emission in the large detuning model. In addition, the interaction time is rather short in the resonant model compared to the large detuning model.Comment: 22 pages, 7 figure

On the deformation of a Barsotti-Tate group over a curve

In this paper, we study deformations of pairs (C,G) where G is a height 2 BT(or BT_n) group over a complete curve on algebraically closed field k of characteristic p. We prove that, if the curve C is a versal deformation of G, then there exists a unique lifting of the pair to the Witt ring W. We apply this result in the case of Shimura curves to obtain a lifting criterion.Comment: 20 pages, no figures. Comments welcom

Role of initial system-bath correlation on coherence trapping

We study the coherence trapping of a qubit correlated initially with a non-Markovian bath in a pure dephasing channel. By considering the initial qubit-bath correlation and the bath spectral density, we find that the initial qubit-bath correlation can lead to a more efficient coherence trapping than that of the initially separable qubit-bath state. The stationary coherence in the long time limit can be maximized by optimizing the parameters of the initially correlated qubit-bath state and the bath spectral density. In addition, the effects of this initial correlation on the maximal evolution speed for the qubit trapped to its stationary coherence state are also explored.Comment: 5 pages,3 figures, welcome to commen