Multiqubit maximally entangled states in the NMR model

Abstract

A single-step operation is proposed to produce multiqubit maximally entangled states in the NMR model. In the scheme, all qubits are initially in the ground state, and one single pulse of a multifrequency coherent magnetic radiation is applied to manipulate simultaneously the ``active states'' that satisfy the resonant conditions while all other ``inactive states'' remain unchanged. An effective Hamiltonian is derived in a generalized rotating frame, which allows us to predict the time evolution of ``active states'' generated by the magnetic pulse. The magnetic pulse parameters, such as frequencies, phases, amplitudes, and duration time, are obtained analytically to implement a Bell state of two qubits and a Greenberger-Horne-Zeilinger state of three qubits. The scheme has been generalized to create an $N$-qubit entangled state. Two rules are found to calculate the magnetic pulse parameters numerically, which are required to realize entangled states for even and odd qubits, respectively. The rules are successfully checked in the cases of $4\ensuremath{\leqslant}N\ensuremath{\leqslant}10$. The relation of $\mathrm{ln}\phantom{\rule{0.3em}{0ex}}{t}_{0}$ and $\mathrm{ln}\phantom{\rule{0.3em}{0ex}}N$ is found to be linear, and the duration time ${t}_{0\phantom{\rule{0.3em}{0ex}}}$is approximately proportional to $\sqrt{N}$.