Entanglement distribution revealed by macroscopic observations

What can we learn about entanglement between individual particles in macroscopic samples by observing only the collective properties of the ensembles? Using only a few experimentally feasible collective properties, we establish an entanglement measure between two samples of spin-1/2 particles (as representatives of two-dimensional quantum systems). This is a tight lower bound for the average entanglement between all pairs of spins in general and is equal to the average entanglement for a certain class of systems. We compute the entanglement measures for explicit examples and show how to generalize the method to more than two samples and multipartite entanglement.

Figure 1

Two contiguous and nonoverlapping spin subsystems $A$ and $B$, each of which contains a large number $n$ of spins. What can we learn about entanglement of a pair $(\alpha ,\beta )$ of spins chosen at random where $\alpha ∊A$ and $\beta ∊B$, if individual spins are experimentally not accessible but only the collective properties of the samples $A$ and $B$? What can we learn about entanglement between $A$ and $B$ from such measurements?

Figure 2

(Color online) Entanglement $E_{ab}$ between two collective spins in the generalized singlet state with an admixture of nonsymmetric noise (11) as a function of spin length $s$ and proportion $p$ of the singlet state in the mixture. The entanglement is nonzero for sufficiently large $p$ and decreases inversely proportionally to $s$.