Entanglement-fluctuation relation for bipartite pure states

We identify subsystem fluctuations (variances) that measure entanglement in an arbitrary bipartite pure state. These fluctuations are of observables that generalize the notion of polarization to an arbitrary $N$-level subsystem. We express this polarization fluctuation in terms of subsystem purity and other entanglement measures. The derived entanglement-fluctuation relation is evaluated for the ground states of a one-dimensional free-fermion gas and the Affleck-Kennedy-Lieb-Tasaki spin chain. Our results provide a framework for experimentally measuring entanglement using Stern-Gerlach–type state selectors.

Figure 1

The linear entropy $S_L$ of a block of $M$ adjacent sites in an infinite periodic chain occupied by noninteracting spinless fermions logarithmically saturates with $M$ to unity. The average fermion density per site is $\nu$. Data points are calculated from the entanglement-fluctuation relation (31) and the smooth curves are obtained from Fisher-Hartwig formulas with $O\left(M^0\right)$ corrections [34, 35, 36, 37].