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arXiv:quant-ph/9709025 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

NMR Greenberger–Horne–Zeilinger states

Raymond Laflamme, E. Knill, Wojciech Hubert Zurek, Paolo Catasti, S. V. Santhana Mariappan

Published Aug 15, 1998• 139 citations
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Abstract

We describe the creation of a Greenberger–Horne–Zeilinger (GHZ) state of the form (|00} + |111})/✓2 (three maximally entangled quantum bits) using Nuclear Magnetic Resonance (NMR). We have successfully carried out the experiment using the proton and carbon spins of trichloroethylene, and confirmed the result using state tomography. We have thus extended the space of entangled quantum states explored systematically to three quantum bits, an essential step for quantum computation.

DOI
10.1098/rsta.1998.0257

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We describe the creation of a Greenberger–Horne–Zeilinger (GHZ) state of the form (|00} + |111})/✓2 (three maximally entangled quantum bits) using Nuclear Magnetic Resonance (NMR). We have successfully carried out the experiment using the proton and carbon spins of trichloroethylene, and confirmed the result using state tomography. We have thus extended the space of entangled quantum states explored systematically to three quantum bits, an essential step for quantum computation.
A scheme is proposed to implement microscopic analogs of the Greenberger-Horne-Zeilinger experiment, as described by Lloyd, on a nuclear magnetic resonance (NMR) quantum computer. This scheme includes how to prepare the effective pure spin state |0000〉 (a state with all the spins pointing along the direction of the magnetic field) from a state at thermal equilibrium, how to transform it into states of the form, e.g., $(|000〉\ensuremath{-}|111〉)|0〉$ (which comprises the first three spins in an entangled state and the fourth register-spin), and how to measure the eigenvalue of the operator product like ${\ensuremath{\sigma}}_{x}^{1}{\ensuremath{\sigma}}_{y}^{2}{\ensuremath{\sigma}}_{y}^{3}$ for such three-spin entangled states, where ${\ensuremath{\sigma}}^{i}$ is the Pauli matrix for the ithi\mathrm{th}ith spin, and store the result on the fourth spin. Also proposed is a general method to implement the controlled-not gate, an essential gate for any quantum algorithm, on n-qubit NMR quantum computers with n> 4.n>~4.n> 4.
The Greenberger-Horne-Zeilinger (GHZ) effect provides an example of quantum correlations that cannot be explained by classical local hidden variables. This paper reports on the experimental realization of GHZ correlations using nuclear magnetic resonance (NMR). The NMR experiment differs from the originally proposed GHZ experiment in several ways: it is performed on mixed states rather than pure states, and instead of being widely separated, the spins on which it is performed are all located in the same molecule. As a result, the NMR version of the GHZ experiment cannot entirely rule out classical local hidden variables. It nonetheless provides an unambiguous demonstration of the ``paradoxical'' GHZ correlations, and confirms the predictions of quantum mechanics.
Abstract This work provides a simple approximate method of creating a Greenberger–Horne–Zeilinger (GHZ) state with high fidelity, using nuclear magnetic resonance (NMR) techniques, and reports on the decoherence process for these GHZ states, which is observed with a pair of encoder and decoder quantum circuits. The prepared GHZ states are verified by measuring the quantum correlations, which are well known as GHZ correlations. The results of the decoherence experiments, show that the GHZ states decay faster than bipartite entangled states such as Bell states by a factor of 1.3, and it appears that two different decoherence processes exist. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
We experimentally explore the state space of three qubits on a nuclear magnetic resonance (NMR) quantum-information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a nontrivial combination of Greenberger-Horne-Zeilinger (GHZ) and WWW-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and WWW states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and WWW states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the WWW state, we demonstrate this method of reconstruction by comparing it to the directly tomographed three-qubit state.
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 NNN-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 lnt0\mathrm{ln}\phantom{\rule{0.3em}{0ex}}{t}_{0}lnt0​ and lnN\mathrm{ln}\phantom{\rule{0.3em}{0ex}}NlnN is found to be linear, and the duration time t0{t}_{0\phantom{\rule{0.3em}{0ex}}}t0​is approximately proportional to N\sqrt{N}N​.