Many copies may be required for entanglement distillation

Abstract

A mixed quantum state shared between two parties is said to be distillable if, by means of a protocol involving only local quantum operations and classical communication, the two parties can transform some number of copies of that state into a single shared pair of qubits having high fidelity with a maximally entangled state state. In this paper it is proved that there exist states that are distillable, but for which an arbitrarily large number of copies is required before any distillation procedure can produce a shared pair of qubits with even a small amount of entanglement. Specifically, for every positive integer n there exists a state that is distillable, but given n or fewer copies of that state every distillation procedure outputting a single shared pair of qubits will output those qubits in a separable state. Essentially all previous examples of states proved to be distillable were such that some distillation procedure could output an entangled pair of qubits given a single copy of the state in question.