Upper bound on the region of separable states near the maximally mixed state

A lower bound on the amount of noise that must be added to a GHZ-like entan gled state to make it separable (also called the random robustness) is foun d using the transposition condition. The bound is applicable to arbitrary n umbers of subsystems, and dimensions of Hilbert space, and is shown to be e xact for qubits. The new bound is compared with previous such bounds on thi s quantity, and found to be stronger in all cases. It implies that increasi ng the number of subsystems, rather than increasing their Hilbert space dim ension, is a more effective way of increasing entanglement. An explicit dec omposition into an ensemble of separable states, when the state is not enta ngled, is given for the case of qubits.