Multipartite Entanglement Witnesses

We derive a set of algebraic equations, the so-called multipartite separability eigenvalue equations. Based on their solutions, we introduce a universal method for the construction of multipartite entanglement witnesses. We witness multipartite entanglement of ${10}^3$ coupled quantum oscillators, by solving our basic equations analytically. This clearly demonstrates the feasibility of our method for studying ultrahigh orders of multipartite entanglement in complex quantum systems.

Figure 1

The entanglement test in Eq. (4) for a $W$ state mixed with white noise, $\widehat{\rho }=p(1/8)\widehat{1}+(1-p)|{\psi }_{\mathrm{W}}⟩⟨{\psi }_{\mathrm{W}}|$ ${\lambda }_1={\lambda }_2={\lambda }_3=(1/\sqrt{3})$, is plotted for $0\le p\le 1$. The boundary for partial or full separability is $4/9$ or $2/3$, respectively. The expectation value $⟨\widehat{L}⟩$ exceeds the boundary for full or partial separability as long as the mixing parameter is $p<(40/63)$ or $p<(8/21)$, respectively.

Figure 2

The expectation value $⟨\widehat{L}⟩$ for a GHZ state mixed with white noise, $\widehat{\rho }=p(1/8)\widehat{1}+(1-p)|{\psi }_{\mathrm{GHZ}}⟩⟨{\psi }_{\mathrm{GHZ}}|$ [${\kappa }_0={\kappa }_1=(1/\sqrt{2})$], is plotted. The lower boundary for partial and full entanglement is $1/2$. The expectation value $⟨\widehat{L}⟩$ exceeds this boundary—which automatically implies genuine entanglement—as long as the mixing parameter is $p<(8/14)$.

Figure 3

The energy difference $\Delta =(⟨\widehat{H}⟩-E[\mathcal{I}])/\mathbf{(}(3/2)\hslash \omega \mathbf{)}$ between quantum states and the ground state of the system is plotted depending on the interaction strength. The boundary ${\Delta }_{1000}$ corresponds to the minimal attainable energy for fully separable states. All quantum states having an energy below ${\Delta }_5$ ($N_1=\cdots =N_5=200$) reveal entanglement between more than 200 oscillators. The quantum states in the range below ${\Delta }_2$ ($N_1=N_2=500$) exhibit entanglement distributed over more than 500 subsystems.