Let $\mathcal{A}$ be a standard operator algebra on a complex Hilbert space H of dimension greater than 2. By invariants of certain functional values of operator products, we characterize some surjective maps on $\mathcal{A}$. Furthermore, several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as, for example, k-numerical radius $(k⩾1)$, operator norm, Ky Fan k-norm, Schatten p-norm $(1⩽p<\infty )$, and so on.

In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain indecomposable optimal witnesses that have no spanning property. These also provide examples that support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).

The question of under what conditions different witnesses (e.g., W1,W2) may detect some common entangled states [i.e., there exists some state $\rho${} so that Tr(W1$\rho${})<0 and Tr(W2$\rho${})0. Second, W2 can detect more entangled states than W1 can if and only if W1=aW2+D for some number a>0 and a positive operator D. As an application, some characterizations of the optimal witnesses are given and some structural properties of the decomposable optimal witnesses are presented.

In the last decade, much attention has been focused on examining the nonlocality of various quantum networks, which are fundamental for long-distance quantum communications. In this paper, we consider the nonlocality of any forked tree-shaped network, where each node, respectively, shares arbitrary number of bipartite sources with other nodes in the next “layer”. The Bell-type inequalities for such quantum networks are obtained, which are, respectively, satisfied by all $(t_n-1)$-local correlations and all local correlations, where $t_n$ denotes the total number of nodes in the network. The maximal quantum violations of these inequalities and the robustness to noise in these networks are also discussed. Our network can be seen as a generalization of some known quantum networks.