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 ${E}_{1}({\ensuremath{\rho}}_{L})=(c∕6)\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L\ensuremath{-}(c∕6)({\ensuremath{\pi}}^{2}∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L)+O(1∕L)$, 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.