Metrological Detection of Multipartite Entanglement from Young Diagrams

We characterize metrologically useful multipartite entanglement by representing partitions with Young diagrams. We derive entanglement witnesses that are sensitive to the shape of Young diagrams and show that Dyson’s rank acts as a resource for quantum metrology. Common quantifiers, such as the entanglement depth and $k$-separability are contained in this approach as the diagram’s width and height. Our methods are experimentally accessible in a wide range of atomic systems, as we illustrate by analyzing published data on the quantum Fisher information and spin-squeezing coefficients.

Figure 1

Representation of multipartite entanglement using Young diagrams. In this example, a system of $N=7$ particles is separable in a partition into $h=3$ subsets and contains an entanglement depth of $w=4$ particles. These quantities correspond to height $h$ and width $w$ of the associated Young diagram. Dyson’s rank $r=w-h$ combines both pieces of information.

Figure 2

Maximal metrological sensitivity for $(w,h)$-separable states as a function of $w$ and $h$. The sensitivity limits are given for (a) the quantum Fisher information $F_Q[\widehat{\rho },{\widehat{J}}_{\mathbf{n}}]$ and (b) the spin-squeezing parameter ${\xi }^2$. By ignoring either $w$ or $h$, we obtain bounds for $w$-producible (red projection) and $h$-separable states (green projection), respectively. A finer projection is given by Dyson’s rank $r=w-h$ [25].

Figure 3

Tuples $(w,h)$ whose separability can be excluded from the experimentally extracted value of $F_Q\ge 40.4$ for $N=14$ parties in Ref. [26] are highlighted. The red tuples (a) are excluded from the entanglement depth (4), the green tuples (b) from $h$-separability (5) and the yellow tuples (c) from Dyson’s rank (6). The gray tuples (d) are found inseparable by Eq. (2), that uses the full information on $(w,h)$ [25]. The results are summarized in the top-view plot (e), where each square represents one tuple $(w,h)$ and example partitions with given width $w$ and height $h$ are illustrated.

Figure 4

(a) Analysis of the multipartite entanglement structure [same as in Fig. 3] for $N=127$ parties based on the measurement of $F_Q[\widehat{\rho },{\widehat{J}}_z]\ge 266.7$ reported in Ref. [29]. (b) Same analysis based on the spin-squeezing coefficient (7) for a measurement of ${\xi }^2\le -4.5\mathrm{dB}$ with $N=470$ particles [17].

Figure 5

Dyson’s rank $r$ as a function of $N$ extracted from published experimental data on the spin-squeezing coefficient ${\xi }^2$ (squares) or the quantum Fisher information $F_Q$ (circles) based the results reported in Refs. [8, 17, 26, 27, 28, 29, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. The quantity $r+N$ ranges between 1 for fully separable states (black line) and $2N-1$ for genuine $N$-partite entangled states (orange line).