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It is well-known that maximally entangled states such as theGreenberger-Horne-Zeilinger (GHZ) states, with the Bell states as the simplestexamples, are widely exploited in quantum information and computation. We studythe application of such maximally entangled states from the viewpoint of theGHZ transform, which is a unitary basis transformation from the product statesto the GHZ states. The algebraic structure of the GHZ transform is made clearand representative examples for it are verified as multi-qubit Clifford gates.In this paper, we focus on the Bell transform as the simplest example of theGHZ transform and apply it to the reformulation of quantum circuit model ofteleportation and the reformulation of the fault-tolerant construction ofsingle-qubit gates and two-qubit gates in teleportation-based quantumcomputation. We clearly show that there exists a natural algebraic structurecalled the teleportation operator in terms of the Bell transform to catchessential points of quantum teleportation, and hence we expect that there wouldalso exist interesting algebraic structures in terms of the GHZ transform toplay important roles in quantum information and computation.

GHZ transform (I): Bell transform and quantum teleportation