2D Projective Complex Space, Spin

[msumm21]

Just reviewing some QM again and I think I'm forgetting something basic. Just consider a qubit with basis {0, 1}. On the one hand I thought 0 and -0 are NOT the same state as demonstrated in interference experiments, but on the other hand the literature seems to say the state space is projective 2D complex Hilbert space and that cS=S for any state S and complex scalar c.

 

[Strilanc]

The difference between $+\left| 0 \right\rangle$ and $-\left| 0 \right\rangle$ matters when the qubit is in superposition. For example, $\frac{1}{\sqrt{2}} \left| 1 \right\rangle + \frac{1}{\sqrt{2}} \left| 0 \right\rangle$ is orthogonal to $\frac{1}{\sqrt{2}} \left| 1 \right\rangle - \frac{1}{\sqrt{2}} \left| 0 \right\rangle$.

The phase of individual states also matters in the context of operations, where the relative phase with other possible outputs is relevant. A 180 degree rotation around the X axis of the bloch sphere sends $\left| 0 \right\rangle$ to $\left| 1 \right\rangle$ and $\left| 1 \right\rangle$ to $\left| 0 \right\rangle$. If you instead send $\left| 1 \right\rangle$ to $-\left| 0 \right\rangle$, you're rotating around the Y axis.

(I used to think that adding a global phase factor to an operation had no effect. This is technically true... until I modified said operation to be controlled by whether or not another qubit was on, so that phase factor only applied in some cases and was suddenly a relative phase factor making my circuit not work.)

 

[kith]

[...] the literature seems to say the state space is projective 2D complex Hilbert space and that cS=S for any state S and complex scalar c.

Wether a system is in state |a> or in state c|a> is undecideable because the Born rule predicts the same probabilities for both. This is the sense in which the states can be considered to be "the same". This doesn't mean that they are algebraically the same.

 

[msumm21]

So if we take two electrons both in the same state 0 then put one in a uniform magnetic field (oriented orthogonal to the direction of the spin) long enough to rotate the state to -0 (half the time required for full state rotation) there's no subsequent experiment we can do afterwards to determine that this system state is different from the case in which neither was in the magnetic field? Was Strilanc saying that we only make these distinctions when analyzing a single electron evolving through a system e.g. the experiment shown at the start of this:http://www-inst.eecs.berkeley.edu/~cs191/fa08/lectures/lecture14_fa08.pdf where the shift to -0 definitely made a difference in that context.

 

[Strilanc]

So if we take two electrons both in the same state 0 then put one in a uniform magnetic field (oriented orthogonal to the direction of the spin) long enough to rotate the state to -0 (half the time required for full state rotation) there's no subsequent experiment we can do afterwards to determine that this system state is different from the case in which neither was in the magnetic field? Was Strilanc saying that we only make these distinctions when analyzing a single electron evolving through a system e.g. the experiment shown at the start of this:http://www-inst.eecs.berkeley.edu/~cs191/fa08/lectures/lecture14_fa08.pdf where the shift to -0 definitely made a difference in that context.

If the electron's path was in superposition, so only one branch went through the magnetic field, then you could tell when recombining the paths (I have no idea if that's easy to do experimentally).

But if you unconditionally apply the phase factor in all possible branches, then no it's not detectable.