Resolution-enhanced entanglement detection

We formulate a general family of entanglement criteria for multipartite states on arbitrary Hilbert spaces. Fisher information criteria compare the sensitivity to unitary rotations with the variances of suitable local observables. Generalized squeezing-type criteria provide lower bounds that are less stringent but require only measurements of second moments. The enhancement due to local access to the individual subsystems is studied in detail for the case of $N$ spin-$1/2$ particles. The discussed techniques can be readily implemented in current experiments with trapped ions in Paul traps and neutral atoms in optical lattices.

Figure 1

Schematic hierarchy of locally and globally optimized coefficients based on the Fisher information or spin squeezing, with or without comparison to local variances. For separable states, the values of all these coefficients are bounded by 1 (dashed horizontal line). The solid lines connect a more efficient entanglement coefficient (above) to a lower bound (below). The dotted lines indicate that all entangled states that are detected by the less efficient coefficient (below) will also be detected by the more efficient one (above), while they may not always provide quantitative bounds for each other. The letters allow identification of the relation with their statement in the text: A, Eq. (2); B, Eq. (37); C, Eq. (47); D, Eq. (40); E, Eq. (48); F, Eq. (46); G, Eq. (38); H, Eq. (41); I, Eq. (45); J, Eq. (44); and K, Eq. (39).

Figure 2

Coefficients $f_l$ and $f_G$ for the noisy “twisted” GHZ states ${\widehat{\rho }}^t\left(p\right)$, Eq. (63), with (a) $N=3$, (b) $N=6$, and (c) $N=9$. In the gray shaded areas (where $f_l>1$, while $f_G\le 1$), the entanglement of the state can be revealed only by local access to the parties. The other coefficients either coincide with those already plotted, i.e., $f_L\equiv f_L^V\equiv f_l$ and $f_G^V\equiv f_G$, or vanish, ${\xi }_l^{-2}\equiv {\xi }_G^{-2}\equiv {\left({\xi }_G^V\right)}^{-2}\equiv 0$, for all $p$.

Figure 3

Entanglement coefficients for the states (64) for $N=3$ as a function of $p$. The squeezing coefficients never exceed the separability limit 1. In contrast, the Fisher densities are able to reveal the state's entanglement. However, local access is required in a finite range of $0.6\lesssim p\lesssim 0.9$.

Figure 4

Entanglement coefficients for the one-axis twisting dynamics, governed by $H_{0,0}$ [Eq. (66)] with an asymmetric initial state (67) and $N=8$ spins. We have $f_L\equiv f_L^V\equiv f_l$ at all times, but only the latter is shown in the plot. The horizontal dash-dotted line at value 1 indicates the separability bound.

Figure 5

Evolution of the entanglement coefficients for the Ising model with transverse field $B=J_0$ and long-range interactions $\alpha =0.2$, including realistic noise processes described by Eq. (68), for $N=8$ spins. The separability limit is indicated by a dash-dotted line at value 1. The lower panel shows a magnified display of the evolution at initial times. The complex dynamics close to the quantum phase transition together with the asymmetric initial condition (67) opens up the full spectrum of entanglement coefficients, summarized in Table 1. The relations displayed in Fig. 1 can be observed, in particular the advantage of local coefficients over global ones. Notice also that at early times the local spin squeezing ${\xi }_l$ is a stronger entanglement witness than the global Fisher information $f_G$.