State interrogation in nuclear magnetic resonance quantum-information processing

Reconstruction of a reduced density operator for weakly coupled systems of spins 1/2 from fits to nuclear magnetic resonance spectra is described in detail. Particular emphasis is placed on data treatment procedures that specify fewer than the 3(n) complete spectra that are implicitly prescribed in published references to state tomography on n-spin systems. It is shown that if the density operator is expanded in the so-called product-operator basis, it is always possible to estimate a desired coefficient in the expansion by measuring a single spectral multiplet. This simple observation can substantially reduce the experimental effort required for either complete density-matrix reconstruction or estimation of subsets of the coefficients in the product-operator expansion. A simple iterative algorithm can be used to produce reduced measurement procedures for experiments involving small numbers of qubits.