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The Journal of Chemical Physics

Entangled electron and nuclear spin states in N15@C60: Density matrix tomography

W. Scherer, Michael Mehring

Published Jan 24, 2008• 32 citations
Publisher

Abstract

Procedures of the preparation and detection of entangled electron-nuclear spin states in (15)N@C(60) by combining electron spin resonance and electron nuclear double resonance pulse techniques are presented. A quantitative evaluation of the complete density matrix is obtained by a special density matrix tomography. All four Bell states of a two qubit subsystem were analyzed and experimental decoherence times are presented. In addition, we estimate a quantum critical temperature of T(q)=7.76 K for this system at an electron spin resonance frequency of 95 GHz.

DOI
10.1063/1.2819310

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Procedures of the preparation and detection of entangled electron-nuclear spin states in (15)N@C(60) by combining electron spin resonance and electron nuclear double resonance pulse techniques are presented. A quantitative evaluation of the complete density matrix is obtained by a special density matrix tomography. All four Bell states of a two qubit subsystem were analyzed and experimental decoherence times are presented. In addition, we estimate a quantum critical temperature of T(q)=7.76 K for this system at an electron spin resonance frequency of 95 GHz.
We have prepared combined electron and nuclear spin pseudoentangled states ${\ensuremath{\Psi}}_{27}^{\ifmmode\pm\else\textpm\fi{}}$ and ${\ensuremath{\Phi}}_{18}^{\ifmmode\pm\else\textpm\fi{}}$ out of the total number of eight quantum states in the multilevel quantum system of a nitrogen atom with electron spin 3/23/23/2 and nuclear spin 1/21/21/2 encaged in the endohedral fullerene 15N@C60^{15}\mathrm{N}@{\mathrm{C}}_{60}15N@C60​. Density matrix tomography was applied to verify the degree of entanglement.
Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.
We perform quantum state tomography (QST) in a three-qubit system consisting of strongly coupled nuclear spins, known in the NMR literature as ABXABXABX systems. We find that the number of experiments necessary to perform QST in such systems can be reduced with respect to those containing three qubits weakly coupled, which reduces the experimental effort required for the complete density-matrix reconstruction. To test the procedure we implement the full protocol for quantum teleportation. The tomographic results demonstrate that the density matrix can be reconstructed with fewer operations and high fidelity.
A method for performing quantum state tomography for quadrupole nuclei is presented in this paper. First, it is shown that upon appropriate phase cycling, the nuclear-magnetic-resonance (NMR) intensities of quadrupole nuclei depend only on diagonal elements of the density-matrix. Thus, a method for obtaining the density-matrix elements, which consists of dragging off-diagonal elements into the main diagonal using fine phase-controlled selective radio-frequency pulses, was derived. The use of the method is exemplified through 23Na^{23}\mathrm{Na}23Na NMR (nuclear spin I=3∕2I=3∕2I=3∕2) in a lyotropic liquid crystal at room temperature, in three applications: (a) the tomography of pseudopure states, (b) the tomography of the quadrupole free evolution of the density matrix, and (c) the unitary state evolution of each qubit in the system over the Bloch sphere upon the application of a Hadamard gate. Further applications in the context of pure NMR and in the context of quantum information processing, as well as generalizations for higher spins, are discussed.