  Studies of entanglement in many-particle systems suggest that most quantum
critical ground states have infinitely more entanglement than non-critical
states. Standard algorithms for one-dimensional many-particle systems construct
model states with limited entanglement, which are a worse approximation to
quantum critical states than to others. We give a quantitative theory of
previously observed scaling behavior resulting from finite entanglement at
quantum criticality: the scaling theory of finite entanglement is only
superficially similar to finite-size scaling, and has a different physical
origin. We find that finite-entanglement scaling is governed not by the scaling
dimension of an operator but by the "central charge" of the critical point,
which counts its universal degrees of freedom. An important ingredient is the
recently obtained universal distribution of density-matrix eigenvalues at a
critical point\cite{calabrese1}. The parameter-free theory is checked against
numerical scaling at several quantum critical points.
