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? 
 A: I am not sure what you might be after, but I'll assume you want the phase-space analog of the Trace distance of two hermitian, normalized density matrices σ and ρ, that is, half the sum of the absolute values of the eigenvalues $\lambda_i$ of their (hermitian) difference, 
$$
T(\rho,\sigma) = \tfrac{1}{2} \operatorname{Tr} \left[ \sqrt{(\rho-\sigma)^2} \right] = \tfrac{1}{2} \sum_i | \lambda_i |   ~.
$$ 
In phase space, via the Wigner map, the density matrix transforms into (real, normalized) Wigner functions, and Hilbert space traces into phase space integrals. 
Thus you end up with something like 
$$
T(f,g)=\tfrac{1}{2} \int dx dp ~\mid (f(x,p)-g(x,p))\mid_\star ~,
$$
where $\mid f\mid _\star$  may be defined as, for instance,  $\tanh_\star(kf) \star f$, with k a positive number, as $k\to \infty$.
For example, the trace distance between the ground state and the first excited state of the quantum harmonic oscillator is something like
$$
T(f_0,f_1)=\frac{1}{\pi \hbar} \int dx dp ~\mid  e^{-(x^2+p^2)/\hbar} ( 1-\frac{x^2+p^2}{\hbar}) \mid _\star  = 1~.
$$
In Hilbert/Fock space, the result is trivial, as $\rho=|0\rangle \langle 0|$,  $\sigma=|1\rangle \langle 1|$, and hence T=(1+1)/2=1. 
In phase space, felicitously, the above integral is computable through a fundamental property of different star-genfunctions of a hamiltonian, their star-orthonormality,  $~f_a\star f_b= \frac{1}{h} \delta_{a,b}~ f_a$, ( Lemma 4, eqn (25) of CTQMPS ). 
Consequently,
$$
\tanh_\star k(f_0-f_1) ~\star f_0= \tanh ( k/h) ~ f_0,
$$
$$
\tanh_\star k(f_0-f_1) ~\star f_1= \tanh(- k/h) ~ f_1,
$$
So the large k limit yields $\mid f_0-f_1\mid_\star=f_0 +f_1$, and since the stargenstates are normalized, the trace distance reduces to its maximum value,
$$
T=\tfrac{1}{2}\int dx dp (f_0+f_1)=(1+1)/2=1.
$$
References:
Quantum Computation and Quantum Information, Michael A Nielsen  & Isaac L Chuang, Cambridge University Press; 10th Anniversary edition (2011) ISBN-13: 978-1107002173
A Concise Treatise on Quantum Mechanics in Phase Space, Thomas L Curtright, David B Fairlie, Cosmas K Zachos, World Scientific (2014) ISBN-13: 978-9814520430

Edits: A different, but related distance measure in Hilbert space is Heydari's, Section 4.4. It is less direct to evaluate, but can be translated to phase space, if needs be.
A very different distance measure Wigner-transforms the 
f-g-asymmetric distance of Mandilara,Karpov & Cerf, PhysRev A79 (2009) 062302, namely, $\quad \int\! dxdp (f-g)^2 ~/(2\int\! dxdp ~f^2)$. 
For the above oscillator example, this evaluates to 1, by inspection.
