Geometric measure of entanglement
The geometric measure of entanglement is a means to quantify the entanglement in a multi-partite system.
For a system consisting of subsystems, the full Hilbert space is a tensor product of those of the subsystems, i.e.,. But for a generic state, it is impossible to write it as a tensor product state. That is, it is impossible to write it in the form of, with. This implies the existence of entanglement between the subsystems.
The geometric measure of entanglement in is then quantified by the minimum of
with respect to all the separable states
with.
This approach works for distinguishable particles or the spin systems. For identical or indistinguishable fermions or bosons, the full Hilbert space is not the tensor product of those of each individual particle. Therefore, a simple modification is necessary. For example, for identical fermions, since the full wave function is now completely anti-symmetric, so is required for. This means, the taken to approximate should be a Slater determinant wave function.
