Monte Carlo (MC) lattice simulation studies have been performed for compressible polymer-polymer interfaces showing a density dip at the interface of an immiscible polymer pair. The dependence of the interfacial properties on the cross interaction energy and pressure is investigated. The simulations are compared with theoretical calculations. Therefore a theory is developed using the lattice fluid (LF) theory applied to polymer interfaces according to Helfand's ideas, using anisotropy factors (probabilities) for the polymer bond directions to account for the inhomogeneity at the interface. The free energy is expressed in terms of these anisotropy factors, polymer densities, and segment-segment contact energies. It is minimized under a number of constraints using the technique of Lagrange multipliers. Between simulations and theory a gratifying qualitative agreement is found. Systems asymmetric in density can be treated, which is important for application to real systems. Application of Helfand's lattice approach is not restricted to broad interfaces and therefore presents a good alternative to gradient term interface theories.