Correlations in local measurements and entanglement in many-body systems

While entanglement plays an important role in characterizing quantum many-body systems, it is hardly possible to directly access many-body entanglement in real experiments. In this paper, we study how bipartite entanglement of many-body states is manifested in the correlation of local measurement outcomes. In particular, we consider a measure of correlation defined as the statistical distance between the joint probability distribution of local measurement outcomes and the product of its marginal distributions. Various bounds of this measure are obtained and several examples of many-body states are considered as a testbed for the measure. We also generalize the framework to the case of imprecise measurement and argue that the considered measure is related to the concept of quantum macroscopicity.

Figure 1

CLM in $z$ directions, optimal CLM for collective spin measurements, and linear entropy of a subsystem, obtained for Haar random states. For each $N$, ${10}^3$ random states were taken and the results were averaged. The green dots represent the analytic values of the average linear entropy. The results show that the averaged CLM for random states decreases whereas the averaged linear entropy increases, indicating that for large $N$, the considered CLM hardly capture the entanglement. The error bars for the linear entropy are invisible as they are too small.

Figure 2

${\mathrm{\Delta }}_D(\widehat{z},\widehat{z})$ (blue curve), linear entropy of a subsystem (red dotted curve), and the upper bound from Proposition t1 (green dashed curve), obtained for spin squeezed states $V_{\mu }{|+\rangle }^{\otimes N}$ as a function of the squeezing strength $\mu$. The system size is $N=200$. The inset shows the entanglement entropy for comparison.

Figure 3

CLMs in $x$ and $z$ directions and linear entropy $1-\mathcal{P}$ of a subsystem for the ground state of the Heisenberg $XXZ$ model. The inset shows the entanglement entropy for comparison.

Figure 4

CLM in $z$ direction for the ground state of the Heisenberg $XXZ$ model for $J_z/J=-1^{+}$, 0, and 1 (from top to bottom) as a function of $N$. Inset: log-log plot of $N$ versus $1-{\mathrm{\Delta }}_D$, which suggests ${\mathrm{\Delta }}_D\approx 1-c{N}^{-\alpha }$ scaling.