Attainable entanglement of unitary transformed thermal states in liquid–state nuclear magnetic resonance with the chemical shift

Recently, Yu, Brown, and Chuang [Phys. Rev. A 71, 032341 (2005)] investigated the entanglement attainable from unitary transformed thermal states in liquid–state nuclear magnetic resonance (NMR). Their research gave an insight into the role of the entanglement in a liquid–state NMR quantum computer. Moreover, they attempted to reveal the role of mixed–state entanglement in quantum computing. However, they assumed that the Zeeman energy of each nuclear spin which corresponds to a qubit takes a common value for all; there is no chemical shift. In this paper, we research a model with the chemical shifts and analytically derive the physical parameter region where unitary transformed thermal states are entangled, by the positive partial transposition (PPT) criterion with respect to any bipartition. We examine the effect of the chemical shifts on the boundary between the separability and the nonseparability, and find it is negligible.