Constraints on the mixing of bipartite states

Abstract

A mixed state may be represented in many different ways as a mixture of pure states ρ=∑pi∣ψi⟩⟨ψi∣. The mixing problem in quantum mechanics asks the characterization of the probability distribution (pi) and the mixed states (ρi) such that ρ=∑piρi for any given mixed state ρ. Some constraints based on eigenvalues of the mixed states are established in uni-party case [see Nielsen, Phys. Rev. A. 63, 052308 (2000), 63, 022144 (2000), Nielsern and Vidal Quantum Inf. Comput. 1 76 (2001)]. We develop some new invariant sets for bipartite mixed states under local unitary operations, which are independent of eigenvalues, and prove some strong constraints based on these invariant sets for the mixing problem in bipartite case. This exhibits a remarkable difference from the uni-party case.