Searching for highly entangled multi-qubit states

We present a simple numerical optimisation procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose (NPT) criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimise this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4 qubit and 5 qubit states discovered by this search. We show that the 4 qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi-Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.