Existence of the Schmidt decomposition for tripartite systems

Abstract

For any bipartite quantum system the Schmidt decomposition allows us to express a pure state vector in terms of a single sum instead of double sums. It is shown that if the partial inner product of a basis (belonging to a Hilbert space of smallest dimension) with the state of the composite system gives a disentangled basis then the Schmidt decomposition exists for tripartite system. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.