The question of under what conditions different witnesses (e.g.,
W1,W2) may detect some common entangled states [i.e., there exists some state $\ensuremath{\rho}$ so that $\mathrm{Tr}({W}_{1}\ensuremath{\rho})<0$ and $\mathrm{Tr}({W}_{2}\ensuremath{\rho})<0$] is answered for both finite-dimensional and infinite-dimensional bipartite systems. Finitely many different witnesses
W1,W2,…,Wn can detect some common entangled states if and only if ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}{W}_{i}$ is still a witness for any nonnegative numbers
d1,d2,…,dn with ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}=1$; they cannot detect any common entangled state if and only if ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}{W}_{i}$ is a positive operator for some nonnegative numbers
c1,c2,…,cn with ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}=1$. For two witnesses
W1 and
W2 more can be said. First,
W1 and
W2 can detect the same set of entangled states if and only if
W1=aW2 for some number
a>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.
Entanglement detection is one of the most fundamental and practical tasks for quantum information processing. The framework of entanglement witnesses provides an experimentally feasible method detecting entangled states. Although It is clear that no entanglement witness per se can detect all entangled states, little is known about how useful a single entanglement witness is. In this work, we show that an entanglement witness can construct another entanglement witness. This means that the same measurement outcomes can be repeatedly applied to constructing different entanglement witnesses.
An entanglement witness is an observable detecting entanglement for a subset of states. We present a framework that makes an entanglement witness twice as powerful due to the general existence of a second (lower) bound, in addition to the (upper) bound of the very definition. This second bound, if non-trivial, is violated by another subset of entangled states. Differently stated, we prove via the structural physical approximation that two witnesses can be compressed into a single one. Consequently, our framework shows that any entanglement witness can be upgraded to a witness
2.0. The generality and its power are demonstrate by applications to bipartite and multipartite qubit/qudit systems.