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 entangled state to make it separable (also called the random robustness) is found using the transposition condition. The bound is applicable to arbitrary numbers of subsystems, and dimensions of Hilbert space, and is shown to be exact for qubits. The new bound is compared with previous such bounds on this quantity, and found to be stronger in all cases. It implies that increasing the number of subsystems, rather than increasing their Hilbert space dimension, is a more effective way of increasing entanglement. An explicit decomposition into an ensemble of separable states, when the state is not entangled, is given for the case of qubits.