Abstract
Critical to the construction of large scale quantum networks, i.e. a quantum internet, is the development of fast algorithms for managing entanglement present in the network. One fundamental building block for a quantum internet is the distribution of Bell pairs between distant nodes in the network. Here we focus on the problem of transforming multipartite entangled states into the tensor product of bipartite Bell pairs between specific nodes using only a certain class of local operations and classical communication. In particular we study the problem of deciding whether a given graph state, and in general a stabilizer state, can be transformed into a set of Bell pairs on specific vertices using only single-qubit Clifford operations, single-qubit Pauli measurements and classical communication. We prove that this problem is NP-Complete.
Highlights
Entanglement takes center stage in the modern understanding of quantum mechanics
In this paper we assume that we already have some existing shared entangled state in a quantum network and we ask the question of whether this state can be transformed into a set of Bell pairs between specific nodes, using only a restricted set of local operations
Our main result is that we prove that the problem of deciding whether a given graph state can be converted into Bell Pairs using only LC + LPM + CC (BellVM) is in general NP-Complete
Summary
Entanglement takes center stage in the modern understanding of quantum mechanics. Apart from its usefulness as a theoretical tool, entanglement can be seen as a resource that can be harnessed for secure communication and many other tasks, see e.g. [1], not achievable by any protocol using only classical communication. In this paper we assume that we already have some existing shared entangled state in a quantum network and we ask the question of whether this state can be transformed into a set of Bell pairs between specific nodes, using only a restricted set of local operations. Examples of such a situation can be found in [2, 3], where an approach is presented of first probabilistically generating a large graph state and transforming this to the desired target state using local operations. In order to prove our results we make heavy use of results in algorithmic graph theory, and we prove new graph theoretical results in the process of proving our main theorem
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