Practical implementations of twirl operations

Twirl operations, which convert impure singlet states into Werner states, play an important role in many schemes for entanglement purification. In this paper we describe strategies for implementing twirl operations, with an emphasis on methods suitable for ensemble quantum information processors such as nuclear magnetic resonance (NMR) quantum computers. We implement our twirl operation on a general two-spin mixed state using liquid-state NMR techniques, demonstrating that we can obtain the singlet Werner state with high fidelity.

Figure 1

Experimental spectra depicting an implementation of an ensemble twirl; the $y$ axis is in arbitrary units, as absolute intensities have no meaning in NMR spectra, but each spectrum is plotted on the same vertical scale. Spectra were obtained using the pulse sequences listed in Table . The four rows correspond to spectra acquired after zero, one, two, or three stages of the twirl sequence, while the left and right columns correspond to direct signal acquisition and acquisition after the selective excitation sequence. The ideal result at the end of the full twirl (last row) is no signal with direct detection (left column) and a pair of antiphase doublets with equal and opposite intensities after a selective excitation sequence (right column).

Figure 2

Spectra from Werner singlet states obtained by twirling impure states with varying singlet fractions; only one component of one antiphase doublet is shown. Successive spectra are obtained by incrementing the variable delay in the preparation sequence, Eq. (12), in steps of $1∕\left(10{\nu }_I\right)=218\mu \mathrm{s}$, which causes the fraction of singlet in the initial state to be modulated, with a period of ten steps. Each spectrum is plotted with the same (arbitrary) vertical scale.