Let $\left(|\uparrow\rangle,|\downarrow\rangle\right)$ and $\left(|\nearrow\rangle,|\swarrow\rangle\right)$ be two bases of the $2$-dimensional Hilbert space $H$.
Can an experiment distinguish between $\frac 1{\sqrt 2} \left(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle\right)$ and $\frac 1{\sqrt 2} \left(|\nearrow\swarrow\rangle - |\swarrow\nearrow\rangle\right)$?
As vectors in the Hilbert space $H\wedge H$, the two clearly coincide (at least up to a phase factor) - simple linear algebra calculation can prove it. But in terms of two $1/2-$spin particles from the Hilbert space $H$, one may think that they are distinct. For example, one may think that one can determine the basis, by an EPR-Bohm experiment. Of course this will not work, since we choose the basis when we choose along which direction of space to measure the spin.
But, is there any known effect in which it matters which basis is used in the singlet state? Is there any kind of "gauge-freedom" associated to the choice of this basis? Are there any theoretical speculations about this?
Update
Seeing the comments (for which I am grateful), I think I should add more clarifications. I let the original question unchanged, and hope this comment can help clarifying what I mean.
There is no difference between $|\psi\rangle$ and $e^{i\vartheta}|\psi\rangle$. Not in theory, but also not in experiment. Two state vectors which differ by a phase factor are undistinguishable (the state is invariant under the action of the group $U(1)$). But if we can't find the phase, we can find the phase differences. Think at interference, or at the Aharonov-Bohm effect.
Now, the singlet state can be seen as being invariant under $SU(2)$. Did anyone try to do something with this "phase"? If it can't be determined, can we at least determine some "smaller" information, similar to the case of the phase?
Can this suggest an experimental test for the Fock space in quantum mechanics?
