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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?


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?

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Absolutely not. They ARE the same state, regardless of how you write it. – Siyuan Ren Sep 3 '12 at 8:36
@Karsus Ren: Thanks. As one can see from my question, I am aware of this: "As vectors in the Hilbert space $H\wedge H$, the two coincide (at least up to a phase factor)- simple linear algebra calculation can prove it.". But being aware of a theoretical explanation doesn't have to stop us from comparing with experiments. – Cristi Stoica Sep 3 '12 at 8:53
@Cristi, I endorse Karsus' answer, of course. There is a more universal source of your confusion. You correctly used the word "represent" in your very question but you don't seem to understand what "represent" means. Representation is a way to talk about something, in your example, about a singlet state. It's a very particular state so it's one thing and can't have different experimental manifestations; it would be a logical oxymoron. If it can be represented differently, the differences are only about the representation - the way how a particular person describes it. – Luboš Motl Sep 3 '12 at 8:56
@CristiStoica: Do you mean to experimentally verify the theoretical prediction that the two states are equivalent in every aspect? – Siyuan Ren Sep 3 '12 at 8:58
Your question is like: Can one distinguish the numbers 1+3 and 2+2 by experiment? It cannot be done as they are just two different ways of referring to the same number 4. – Arnold Neumaier Sep 3 '12 at 9:12

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