Information-theoretic measure of genuine multiqubit entanglement

We consider pure quantum states of $N$ qubits and study the genuine $N$-qubit entanglement that is shared among all the $N$ qubits. We introduce an information-theoretic measure of genuine $N$-qubit entanglement based on bipartite partitions. When $N$ is an even number, this measure is presented in a simple formula, which depends only on the purities of the partially reduced density matrices. It can be easily computed theoretically and measured experimentally. When $N$ is an odd number, the measure can also be obtained in principle.

Figure 1

(Color online) Information diagram for (a) product pure states $\mid \psi {⟩}_{A_1{B}_1}\otimes \mid \psi {⟩}_{A_2{B}_2}$; the bipartite nonlocal information (overlap) between $A=A_1\cup A_2$ and $B=B_1\cup B_2$ is $S_{A\mid B}=I_{A_1{B}_1}+I_{A_2{B}_2}$; (b) two-qubit (top) and three-qubit (bottom) pure states.

Figure 2

(Color online) Genuine $N$-qubit entanglement ${\mathcal{E}}_{12\cdots N}$ for the ground state of the finite transverse field Ising model with $N=4$, 6, 8, and 10.