density operator - How can we define the distance for a pair quantum states in phase space?

In condensed matter physics, we know that two degenerate ferromagnetic ground states $|\uparrow\uparrow ...\uparrow \rangle$ and $|\downarrow\downarrow...\downarrow\rangle$ are far from each other in phase space because it needs lots of operations to convert one to another.

Is there a quantity like "distance" in phase space to describe such a property?