Detection and characterization of multipartite entanglement in optical lattices

We investigate the detection and characterization of entanglement based on the quantum network introduced in Phys. Rev. Lett. 93, 110501 (2004) for different experimental scenarios. We first give a detailed discussion of the ideal scheme where no errors are present and full spatial resolution is available. Then we analyze the implementation of the network in an optical lattice. We find that even without any spatial resolution entanglement can be detected and characterized in various kinds of states including cluster states and macroscopic superposition states. We also study the effects of detection errors and imperfect dynamics on the detection network. For our scheme to be practical these errors have to be on the order of one over the number of investigated lattice sites. Finally, we consider the case of limited spatial resolution and conclude that significant improvement in entanglement detection and characterization compared to having no spatial resolution is only possible if single lattice sites can be resolved.

Figure 1

Effect of varying $\varphi$ on $\mathrm{Tr}\left\{{\rho }_B^2\right\}$ where $B$ is any one atom not at an end (dotted), any two atoms not at ends and with at least two others between them (dashed), any two or more consecutive atoms not including an end (dash-dotted), any one or more consecutive atoms including one end (solid). These purities are independent of $n$. For $n⩾6$ corresponding subsets are, e.g., $B=\left\{2\right\}$, {2, 5}, {2, 3, 4, 5}, {1, 2, 3, 4, 5}, respectively. Hence the state is entangled if the purities as listed above are not in decreasing order, that is, for all $\varphi$ except integer multiples of $2\pi$.

Figure 2

Entanglement detection network acting on two rows I and II of identical qubits. The two rows of $n$ qubits are prepared in the (possibly entangled) state ${\rho }_{1,2,3,\dots ,n}\otimes {\rho }_{1,2,3,\dots ,n}$ and performing the pairwise BSs followed by measuring the lattice site occupancies realizes the entanglement detection network by projecting each operator ${\rho }_j$ into its symmetric and antisymmetric subspaces.

Figure 3

$\overline{\mathrm{Tr}\left\{{\rho }_{\left(k\right)}^2\right\}}$ against $k$ (left) and $P\left(j\right)$ against $j$ (right) for $n=10$. (a) Classically correlated state $\rho =({\mid 0⟩}^n{⟨0\mid }^n+{\mid 1⟩}^n{⟨1\mid }^n)∕2$. (b) Maximally entangled state $\mid {(\gamma =0)}_{10}⟩$. (c) Cluster state $\mid {(\varphi =\pi )}_{10}⟩$. (d) Same as (c) with 10% dephasing decoherence.

Figure 4

Average purities $\overline{\mathrm{Tr}\left\{{\rho }_{\left(k\right)}^2\right\}}$ ($k=1$ dotted curve; $k=2$ dashed curve; $k=3$ dash-dotted curve; $k=7$ solid curve) for the state $\mid {\varphi }_{n=15}⟩$. Since all states are pure we have $\overline{\mathrm{Tr}\left\{{\rho }_{\left(k\right)}^2\right\}}=\overline{\mathrm{Tr}\left\{{\rho }_{(n-k)}^2\right\}}$.

Figure 5

Effect of dephasing noise on the $n=15$ cluster state, $k=1$ (dotted), 2 (short dashed), 8 (long dashed), 14 (dash-dotted), and 15 (solid).

Figure 6

Variance $V_k$ against BS error, for $k=1$ (dotted), 4 (short dashed), 7 (long dashed), 11 (dash-dotted), 15 (solid), and $n=15$. (a) shows the result for the worst case and (b) for a cluster state.

Figure 7

Worst-case variance against $n$ with BS error $q=1∕k$, for $k=2$ (diamonds), 4 (triangles), 7 (hollow squares), $n$ (solid squares). The dotted horizontal line is the analytic bound $V_k⩽\exp \left(4kq\right)$. The solid curves are drawn to guide the eye.

Figure 8

Variance against detection error without spatial resolution using Eq. (33), for $k=1$ (dotted), 4 (short dashed), 7 (long dashed), 11 (dash-dotted), 15 (solid), and $n=15$. (a) shows the result for the worst case and (b) for a cluster state.

Figure 9

Worst-case variance against $n$ with detection error $p=1∕\left(2k\right)$ using least squares, for $k=1$ (diamonds), 2 (triangles), 3 (hollow squares), $n$ (solid squares). The solid curves are drawn to guide the eye.

Figure 10

Variance of $\mathrm{Tr}\left\{{\rho }_B^2\right\}$ as a function of the standard deviation $\sigma$, for $B=\left\{2\right\}$ (dotted), {2,3} (dashed), {1,3} (dash-dotted), {1,2,3} (solid), and $n=4$. (a) and (b) are worst-case variances while (c) and (d) are for a maximally entangled state. (a) and (c) each show four almost coincident curves obtained from Eq. (38) while (b) and (d) are found by the least-squares method.