An orthogonal basis for the hyperbolic hybrid polynomial space

Motivated by the wlde usage of the Tchebyshev basis and Legendre basis in the algebra polynomial space, we construct an orthogonal basis with the properties of the H-Bézier basis In the hyperbolic hybrid polynomial space, which is similar to the Legendre basis and holds remarkable properties. Moreover, we derive the transformation matrices that map the H-Bézler basis and the orthogonal basis forms into each other. An example for approximating the degree reduction of the H- Bézier curves Is sketched to Illustrate the utility of the orthogonal basis.

THE CONSTRUCTION OF ORTHOGONAL WAVELET BASIS ON[0,1] AND NUMERICAL SIMULATION

In this paper, we show the construction of orthogonal wavelet basis on the interval [0, 1],using compactly supportted Daubechies function. Forwardly, we suggest a kind of method to deal with the differential operator in view of numerical analysis and derive the appoximation algorithm of wavelet ba-sis and differential operator, which affects on the basis, to functions belonging to L2 [0, 1 ]. Numerical computation indicate the stability and effectiveness of the algorithm.

An efficient quantum key distribution protocol with orthogonal product states

An efficient quantum key distribution （QKD） protocol with orthogonal product states in the 3 3 Hilbert space is presented. The sender, Alice, disorders the orthogonal product state sequence and sends it to Bob. After Alice has published the matching information of the particle sequence, Bob recovers the correct correspondences and makes an orthogonal measurement on the orthogonal product states to obtain the information sent by Alice. Finally, security analysis is also made.

Scheme for Cloing a Three-Particle GHZ Class State with Assistance

Our concern is to design an assisted-clone scheme which can produce a perfect copy of a three-particle Oreenberger-Horne-Zeilinger （GHZ） class state with a high probability. In the first stage of the protocol, the sender teleports the input state to the receiver by using three EPR pairs as the quantum channel. In the second stage of the protocol, a novel set of mutually orthogonal basis vectors is constructed. With the assistance of the preparer through a three-particle projective measurement under this basis, the perfect copy of an original state can be reestablished by the sender with the probability 1/2. Moreover, the classical communication cost of the scheme is also calculated.

Deterministic assisted clone of an unknown four-particle entangled cluster-type state

Two deterministic schemes are proposed to realize the assisted clone of an unknown four-particle entangled cluster- type state. The schemes include two stages. The first stage requires teleportation via maximal entanglement as the quantum channel. In the second stages of the protocols, two novel sets of mutually orthogonal basis vectors are constructed, With the assistance of the preparer through a four-particle or two-step two-particle projective measurement under these bases, the perfect copy of an original state can be produced. Comparing with the previous protocols which produce the unknown state and its orthogonal complement state at the site of the sender, the proposed schemes generate the unknown state deterministically.

Deterministic Remote State Preparation via the χ State

Two deterministic schemes using the χ state as the entangled channel are put forward to realize the remote preparation of arbitrary two-and three-qubit states. To design the schemes, we construct sets of ingenious measurement bases, which have no restrictions on the coefficients of the prepared state. At variance with the existing schemes via the χ state, the success probabilities of the proposed schemes are greatly improved.

Optimizing scheme for probabilistic remote preparation of a two-qubit state

Remote state preparation is increasingly becoming attractive in recent years, people have already started theoretical and experimental research, and have made valuable research results. Recently, a scheme for probabilistic remote preparation of a general two-qubit state was proposed （Wang Z Y in Quantum Inf Process. 11：1585, 2012））. In this paper, we present a modified scheme for probabilistic remote preparation of a general two-qubit state. To complete the scheme, the new and feasible complete orthogonal basis vectors have been introduced. Compared with the previous schemes, the advantage of our schemes is that the total success probability of remote state preparation will be greatly improved. The probability of success regarding this scheme is calculated in both general and particular cases. The results show that the success probability of remote state preparation can be improved a little. However, in certain special cases, the success probability of preparation can be greatly improved. In special cases, the success probability of preparation can be improved to 1. The security analysis of the scheme is provided in details.