Remarks on the structure of states of composite quantum systems and envariance

Some basic aspects of the theoretical description of composite quantum systems are reconsidered, namely Schmidt decomposition for the states of bi-partite systems and the reduction of density matrices. For both we offer new and comprehensive proofs, under weaker assumptions. Furthermore we present some new concrete applications of envariance for entangled quantum state, reviewing partially Zurek's original proposal. Finally some implications of the general results for the Schmidt decomposition for Zurek's programme are outlined.