Blind quantum computation allows a client without enough quantum technologies to delegate her quantum computation to a remote quantum server, while keeping her input, output and algorithm secure. In this paper, we propose a universal single-server an
d almost-classical-client blind quantum computation protocol. In this protocol, the client interfaces with only one server and the only ability of the client required is to get particles from the trusted center and forward them to the server. We present an analysis of this protocol and modify it to a universal single-server and fully-classical-client blind quantum computation protocol by improving the ability of the trusted center. Based on our protocols and recent works, a new Cloud + Certificate Authority (CA) style is proposed for the blind quantum computation.
We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the fo
rm of $W^{wopt}=sigma-c_{sigma}^{max} I$, where $c_{sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzic{a}g {it et al}, Phys. Rev. A {bf 88}, 010301(R) (2013)], and we examine their geometric properties.
Any bipartite entanglement witness $W$ can be written as $W=c_{sigma}I-sigma$, where $sigma$ is a quantum state, $I$ is the identity matrix, and $c_{sigma}$ is a non-negative number. We present a general method to extend the given entanglement witnes
s to multipartite cases via purification, partial purification, and direct tensor of the quantum state $sigma$. Our methods extend $sigma$ but leave the parameter $c_{sigma}$ untouched. This is very valuable since the parameter is generally not easy to compute.
Based on the general form of entanglement witnesses constructed from separable states, we first show a sufficient condition of violating the structural physical approximation (SPA) conjecture [Phys. Rev. A 78, 062105 (2008)]. Then we discuss the SPA
conjecture for decomposable entanglement witnesses. Moreover, we make geometric illustrations of the connection between entanglement witnesses and the sets of quantum states, separable states, and entangled states comparing with planes and vectors in Euclidean space.
We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=rho-c_{rho} I$, where $rho$ is a den
sity matrix, $c_{rho}$ is a non-negative number related to $rho$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio{l}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=rho-c_{rho}I$ in operation, as well as in theory.
