Unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder inequalities

We present a unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to use it to derive not only already known entanglement criteria, but also criteria which were unknown so far. We systematically investigate its potential and limitations to detect bipartite and multipartite entanglement. Furthermore, we describe how to apply our generalized entanglement detection scheme to find the measurement directions to verify the entanglement of a given state experimentally.

Figure 1

Entanglement detection of ${\varrho }_{\alpha }$ [Eq. (57)] with the help of Eq. (54). $E>0$ indicates entanglement. For the crosses, the measurement basis was optimized for each $\alpha$, whereas for the pluses we used the optimized basis for $\alpha =2$ for all $\alpha$. See the text for further details.