Ja n 20 05 The Existence of Quantum Entanglement Catalysts ∗

Without additional resources, it is often impossible to transform one entangled quantum state into another with local quantum operations and classical communication. Jonathan and Plenio [Phys. Rev. Lett. 83, 3566(1999)] presented an interesting example showing that the presence of another state, called a catalyst, enables such a transformation without changing the catalyst. They also pointed out that in general it is very hard to find an analytical condition under which a catalyst exists. In this paper we study the existence of catalysts for two incomparable quantum states. For the simplest case of 2×2 catalysts for transformations from one 4×4 state to another, a necessary and sufficient condition for existence is found. For the general case, we give an efficient polynomial time algorithm to decide whether a k × k catalyst exists for two n× n incomparable states, where k is treated as a constant. Index Terms — Quantum information, entanglement states, entanglement transformation, entanglement catalysts.