Entanglement Purification and Distillation

Abstract

In quantum mechanics, subsystems of a composite system can exhibit correlations (correlations in quantum mechanics) that are stronger than any classical correlations. Quantum correlations are also called entanglement [1]. A mixed quantum state consisting of two subsystems (i.e. a bipartite state) can be either separable or entangled. It is separable [2] if = i p i | a i a i | ⊗ | b i b i |, with p i being probabilities , and entangled otherwise. Entanglement can be quantified via entanglement measures. Maximally entangled states are pure, and mixing generally decreases en-tanglement. For further reading on entanglement, see [18-20] and general textbooks on quantum information, e.g. [21-23]. In quantum information entanglement is viewed as a resource, see protocols such as quantum teleportation [3], superdense coding [4] or entanglement-based quantum cryptography (quantum communication) [5]. Therefore, one is interested in maximally entangled (pure) quantum states. In a realistic scenario, noise due to interaction with the environment (decoherence) or imperfect gate operations generally reduces both purity and entanglement of a given state. However, if one has several copies of some less than maximally entangled state available, it is possible that the two parties Alice (A) and Bob (B) concentrate or distill the entanglement, by acting locally on their parts of the states (in their corresponding laboratories) and exchanging classical information via a telephone. Thus, by using so-called local operations and classical communication (LOCC) they can create fewer pairs with higher entanglement and higher degree of purity. This process is called entangle-ment purification or entanglement distillation. In this context, two topics are of interest: First, one wants to find distillation protocols that are as efficient as possible. Second, one studies the possibility of distillation. The "distillability problem" is phrased as: given a certain density matrix , is it distillable or not? For pure, but not maximally entangled states, it is possible to increase the entan-glement by "local filtering" [6]. Here Alice and Bob apply certain local operators, and with some probability p arrive at a state with higher entanglement. However, as it is not possible to increase entanglement on average by local operations, with probability 1 − p the resulting state is less entangled than before. The first purification and distillation protocols for mixed states were suggested in [7, 8]. In [7] the given state is first brought by random local rotations into a standard form,