  We investigate the behavior of the moments of the partially transposed
reduced density matrix \rho^{T_2}_A in critical quantum spin chains. Given
subsystem A as union of two blocks, this is the (matrix) transposed of \rho_A
with respect to the degrees of freedom of one of the two. This is also the main
ingredient for constructing the logarithmic negativity. We provide a new
numerical scheme for calculating efficiently all the moments of \rho_A^{T_2}
using classical Monte Carlo simulations. In particular we study several
combinations of the moments which are scale invariant at a critical point.
Their behavior is fully characterized in both the critical Ising and the
anisotropic Heisenberg XXZ chains. For two adjacent blocks we find, in both
models, full agreement with recent CFT calculations. For disjoint ones, in the
Ising chain finite size corrections are non negligible. We demonstrate that
their exponent is the same governing the unusual scaling corrections of the
mutual information between the two blocks. Monte Carlo data fully match the
theoretical CFT prediction only in the asymptotic limit of infinite intervals.
Oppositely, in the Heisenberg chain scaling corrections are smaller and,
already at finite (moderately large) block sizes, Monte Carlo data are in
excellent agreement with the asymptotic CFT result.
