A finite entanglement entropy and the c-theorem

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with nonzero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A, B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F (A, B) is severely constrained by the Poincare symmetry and the mathematical properties of the entropy. In particular, for one component sets in two-dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for (1 + 1)-dimensional QFT. We propose this well-defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem. (C) 2004 Elsevier B.V. All rights reserved.