Separability of very noisy mixed states and implications for NMR Quantum computing

We give a constructive proof that all mixed states of N qubits in a suffici ently small neighborhood of the maximally mixed state are separable (unenta ngled). The construction provides an explicit representation of any such st ate as a mixture of product states. We give upper and lower bounds on the s ize of the neighborhood, which show that its extent decreases exponentially with the number of qubits. The bounds show that no entanglement appears in the physical states at any stage of present NMR experiments. Though this r esult raises questions about NMR quantum computation, further analysis woul d be necessary to assess the power of the general unitary transformations, which are indeed implemented in these experiments, in their action on separ able states.