Assisted optimal state discrimination without entanglement

A fundamental problem in quantum information is to explore the roles of different quantum correlations in a quantum information procedure. Recent work [Phys. Rev. Lett., 107 (2011) 080401] shows that the protocol for assisted optimal state discrimination (AOSD) may be implemented successfully without entanglement, but with another correlation, quantum dissonance. However, both the original work and the extension to discrimination of $d$ states [Phys. Rev. A, 85 (2012) 022328] have only proved that entanglement can be absent in the case with equal a \emph{priori} probabilities. By improving the protocol in [Sci. Rep., 3 (2013) 2134], we investigate this topic in a simple case to discriminate three nonorthogonal states of a qutrit, with positive real overlaps. In our procedure, the entanglement between the qutrit and an auxiliary qubit is found to be completely unnecessary. This result shows that the quantum dissonance may play as a key role in optimal state discrimination assisted by a qubit for more general cases.Comment: 6 pages, 3 figures. Accepted by EPL. We extended the protocol for assisted optimal state discrimination to the case with positive real overlaps, and presented a proof for the absence of entanglemen

Most robust and fragile two-qubit entangled states under depolarizing channels

For a two-qubit system under local depolarizing channels, the most robust and most fragile states are derived for a given concurrence or negativity. For the one-sided channel, the pure states are proved to be the most robust ones, with the aid of the evolution equation for entanglement given by Konrad et al. [Nat. Phys. 4, 99 (2008)]. Based on a generalization of the evolution equation for entanglement, we classify the ansatz states in our investigation by the amount of robustness, and consequently derive the most fragile states. For the two-sided channel, the pure states are the most robust for a fixed concurrence. Under the uniform channel, the most fragile states have the minimal negativity when the concurrence is given in the region [1/2,1]. For a given negativity, the most robust states are the ones with the maximal concurrence, and the most fragile ones are the pure states with minimum of concurrence. When the entanglement approaches zero, the most fragile states under general nonuniform channels tend to the ones in the uniform channel. Influences on robustness by entanglement, degree of mixture, and asymmetry between the two qubits are discussed through numerical calculations. It turns out that the concurrence and negativity are major factors for the robustness. When they are fixed, the impact of the mixedness becomes obvious. In the nonuniform channels, the most fragile states are closely correlated with the asymmetry, while the most robust ones with the degree of mixture.Comment: 10 pages, 9 figs. to appear in Quantum Information & Computation (QIC

Convolutional Neural Networks combined with Runge-Kutta Methods

A convolutional neural network for image classification can be constructed mathematically since it can be regarded as a multi-period dynamical system. In this paper, a novel approach is proposed to construct network models from the dynamical systems view. Since a pre-activation residual network can be deemed an approximation of a time-dependent dynamical system using the forward Euler method, higher order Runge-Kutta methods (RK methods) can be utilized to build network models in order to achieve higher accuracy. The model constructed in such a way is referred to as the Runge-Kutta Convolutional Neural Network (RKNet). RK methods also provide an interpretation of Dense Convolutional Networks (DenseNets) and Convolutional Neural Networks with Alternately Updated Clique (CliqueNets) from the dynamical systems view. The proposed methods are evaluated on benchmark datasets: CIFAR-10/100, SVHN and ImageNet. The experimental results are consistent with the theoretical properties of RK methods and support the dynamical systems interpretation. Moreover, the experimental results show that the RKNets are superior to the state-of-the-art network models on CIFAR-10 and on par on CIFAR-100, SVHN and ImageNet

Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and updating basis functions in selected iterations.Comment: 19 pages, 2 figures, submitted to Journal of Computational and Applied Mathematic