We show that three distant labs A, B and C, having no prior entanglement can establish a shared GHZ state, by mediating two particles which remain separable from each other and from all the other parties throughout the process. The success probability is 1 7. We prove this in a general framework for systematic distribution of entangled states between 2 and more parties with separable states in d dimensions. The proposed method may facilitate the construction of multi-node quantum networks and many other processes which use multi-partite entangled states.Introduction Decades of study of entanglement, from its inception in relation with conceptual framework of the quantum theory [1][2][3], to its recent upsurge as a tool in quantum computation and quantum information science [4] and the inevitable need for its quantification has shown that its various unexplored aspects are still intriguing and can still surprise us. In the history of study of entanglement, the turning point was the discovery that entanglement can be used to teleport quantum states [5], to share secret keys and to do dense coding, all pointing to the fact that this concept is a useful resource, as important as many other physical quantities. Since then it has been shown that entanglement can be manipulated [6], measured [7], distributed [8] and distilled [9]. The distinctive feature of entanglement is that it cannot be created by local action and classical communication (LOCC) [10]. Any attempt for distributing entanglement between two or more distant particles, should involve either a direct interaction between the particles when they are close [11] or should be mediated between them by sending a mediating particle through a quantum channel [12]. This latter method is of course very much vulnerable to noise, since the state of mediating particle is very fragile. This problem becomes very severe if we note that the mediating particle may be entangled with the distant particles.
We investigate noise effects on the performance of entanglement distribution by separable state. We consider a realistic situation in which the mediating particle between two distant nodes of the network goes through a noisy channel. For a large class of noise models we show that the average value of distributed entanglement between two parties is equal to entanglement between particular bipartite partitions of target qubits and exchange qubit in intermediate steps of the protocol. This result is valid for distributing two qubit/qudit and three qubit entangled states. In explicit examples of the noise family, we show that there exists a critical value of noise parameter beyond which distribution of distillable entanglement is not possible. Furthermore, we determine how this critical value increases in terms of Hilbert space dimension, when distributing d-dimensional Bell states.
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