Half the entanglement in critical systems is distillable from a single specimen

We establish a quantitative relationship between the entanglement content of a single quantum chain at a critical point and the corresponding entropy of entanglement. We find that, surprisingly, the leading critical scaling of the single-copy entanglement with respect to any bipartitioning is exactly one-half of the entropy of entanglement, in a general setting of conformal field theory and quasifree systems. Conformal symmetry imposes that the single-copy entanglement scales as $E_1\left({\rho }_L\right)=(c∕6)\ln L-(c∕6)({\pi }^2∕\ln L)+O(1∕L)$, where $L$ is the number of constituents in a block of an infinite chain and $c$ denotes the central charge. This shows that from a single specimen of a critical chain, already half the entanglement can be distilled compared to the rate that is asymptotically available. The result is substantiated by a quantitative analysis for all translationally invariant quantum spin chains corresponding to all isotropic quasifree fermionic models. An example of the $XY$ spin chain shows that away from criticality the above relation is maintained only near the quantum phase transition.

Figure 1

Contour of integration to be taken in case of both the entropy of entanglement and the single-copy entanglement.