Compact set of invariants characterizing graph states of up to eight qubits

The set of entanglement measures proposed by Hein, Eisert, and Briegel for $n$-qubit graph states [Phys. Rev. A 69, 062311 (2004)] fails to distinguish between inequivalent classes under local Clifford operations if $n\ge 7$. On the other hand, the set of invariants proposed by van den Nest, Dehaene, and De Moor (VDD) [Phys. Rev. A 72, 014307 (2005)] distinguishes between inequivalent classes, but contains too many invariants (more than $2\times {10}^{36}$ for $n=7$) to be practical. Here we solve the problem of deciding which entanglement class a graph state of $n\le 8$ qubits belongs to by calculating some of the state’s intrinsic properties. We show that four invariants related to those proposed by VDD are enough for distinguishing between all inequivalent classes with $n\le 8$ qubits.

Figure 1

Graphical effect of local complementation on qubit $i$. Local complementation on qubit $i$ on the graph on the left (right) leads to the graph on the right (left).