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 sufficiently small neighbourhood of the maximally mixed state are separable. The construction allows us to explicitly represent any such state as a mixture of product states. We give upper and lower bounds on the size of the neighbourhood which show in particular that its extent decreases exponentially with the number of spins. We also discuss the implications of the bounds for NMR quantum computing