Quantum-state tomography for quadrupole nuclei and its application on a two-qubit system

Abstract

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 23 Na NMR (nuclear spin I =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.The 23 Na NMR experiments described in this paper were performed using a 9.4 T VARIAN INOVA spectrometer in a lyotropic liquid-crystal system prepared with 35 . 9 wt. % of sodium decyl sulfate (Fluka), 7 . 2 wt. % of decanol (Su-*

Quantum information processing through nuclear magnetic resonance

“…The technique to rebuilt ρ from measured NMR signals is called Quantum State Tomography and was suggested by Vogel and Risken [24], for spin 1/2 systems. For quadrupolar spins, the technique was first reported by Bonk et al [25]. The deviation density matrix (labeled here as ρ) can de considered to have the general form "( Since NMR is not sensitive to the unity matrix of pseudo-pure states, Eq.…”

“…The method described in [25] is based on the fact that the NMR spectra amplitudes are related only to the diagonal elements of ρ: …”

“…where e i j are the elements of the matrix which represent the read-out pulse [25]. The other elements are obtained applying…”

“…More information about the sample preparation and experimental procedures can be found in references [1], [2], [25] and [26].…”

Quantum State Tomography and Quantum Logical Operations in a Three Qubits NMR Quadrupolar System

“…This method was later adapted for homonuclear coupled spins 1/2 [10,11] and also for quadrupolar spin 3/2 systems [12][13][14][15][16][17], where nonselective pulses were replaced by transitionselective RF pulses. The disadvantage of such method is the long time required to perform a selective pulse, producing considerable unwanted evolution of the system during the QST process.…”