Bounds on the entanglability of thermal states in liquid-state nuclear magnetic resonance

Abstract

Theorists have recently shown that the states used in current nuclear magnetic resonance (NMIR) quantum computing experiments are not entangled. Yet it is widely believed that entanglement is a necessary resource in the implementation of quantum algorithms. The apparent contradiction might be resolved by the experimental realization of an entangled NMR state. Designing such an experiment requires us to know whether or not the initial NMR state is entanglable--that is, does there exist a unitary transform that entangles the state? This computational and theoretical thesis explores the entanglability of thermal states in N-[alpha] space where N specifies the number of qubits and [alpha] characterizes the polarization of the thermal state. The thermal state is transformed by the Bell unitary U[sub]b,s and the entanglement of the transformed state is measured by negativity. Here we present numerically generated negativity maps of N-[alpha] space (N [less than or equal to] 12) and explicit negativity formulas for U[sub]b,s-transformed thermal states. We also give a general method that uses the symmetry of a special mixed Bell state family to derive bounds on the entanglement of generic Bell-transformed thermal states. This approach yields analytical bounds on the entanglability of thermal states and gives an upper limit of N [less than or equal to] 20, 054 required to entangle a thermal state under ideal experimental conditions.