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International Journal of Quantum Information

CONSTRUCTING MAXIMALLY MULTIQUBIT ENTANGLED STATE VIA BELL-PAIR STATE

Xin-Wei Zha, Jianxia Qi

Published Jun 15, 2011• 6 citations
Publisher

Abstract

Brown et al. and Borras et al. have explored numerical optimization procedures to search for highly entangled five- and six-qubit states. Here, we give a relation between the genuine multiqubit entangled state and Bell state for 4-, 5-, and 6-qubit systems. And the general method to construct maximally multipartite entangled state is also presented.

DOI
10.1142/s0219749911007976

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Brown et al. and Borras et al. have explored numerical optimization procedures to search for highly entangled five- and six-qubit states. Here, we give a relation between the genuine multiqubit entangled state and Bell state for 4-, 5-, and 6-qubit systems. And the general method to construct maximally multipartite entangled state is also presented.
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Recently, Paolo Facchi et al. [Phys. Rev. A. 77, 060304 (R) (2008)] introduced the notion of maximally multipartite entangled states of n qubits. Here, we give a criterion for faithful controlled teleportation of an arbitrary two-qubit state via a five-qubit entangled state and obtain the general relation between the genuine five-qubit entangled state and controlled teleportation. This criterion can be extended to teleportation of an arbitrary N-qubit state using 2N+1-qubit entangled state. Furthermore, we study the optimal match of measuring basis and quantum channel.
Abstract The notion of a maximally entangled state depends on the measure used to give rise to a rich envision of multipartite state. Recently, Che et al presented a recurrence relation of multi-qubit states. According to this inspiration, we present an expanded recurrence relation of the maximally multipartite state. Using this expanded recurrence relation and Bell state, some new forms of the maximally four-, six-, eight-qubits states are presented. The construct of the maximally multipartite state through recurrence relationships is of great potential.
Since the first quantum secret sharing protocol based on Greenberger–Horne–Zeilinger (GHZ) states was presented by Hillery et al, dozens of quantum secret sharing protocols based on different entangled states (e.g. Bell states, d-level Bell states and GHZ states, etc) and product states have been developed. In this paper, we present a quantum secret sharing protocol, in which the genuinely maximally entangled six-qubit states, which was found recently by Borras et al, are used. An ordinary entangled state such as a Bell state or a GHZ state, etc can be used to share at most only one bit. However, according to our protocol, Alice can share three bits among three parties by a six-qubit state, which shows that our protocol is more efficient than any previous protocol.
We present a simple numerical optimization procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimize this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4-qubit and 5-qubit states discovered by this search. We show that the 4-qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi–Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.