- #1
Kreizhn
- 743
- 1
I'm having a bit of a brain fart here, so hopefully someone can help.
Consider a closed, two-level quantum system. We know we can describe pure states as
[tex] \alpha |0\rangle + \beta |1 \rangle [/tex]
for some orthonormal basis [itex] |0\rangle, |1 \rangle [/itex]. The normalization conditions means we can associate this space with the unit 3-sphere
[tex] S^3 = \left\{ (z_1,z_2) \in \mathbb C^2 : |z_1|^2 + |z_2|^2 = 1 \right\} [/tex]
Now on the other hand, we also use the Bloch sphere to describe such states, where geometrically the Bloch sphere is just [itex] \mathbb{PC}^1[/itex] the complex projective line.
So I'm wondering, how do we map [itex] S^3 \to \mathbb{PC}^1 [/itex] quantum mechanically? What does this map look like? Is this occurring because we're throwing away the global phase in our transition to the Bloch sphere?
Something is telling me that Hopf fibrations are important here, but I just can't seem to put two and two together.
Consider a closed, two-level quantum system. We know we can describe pure states as
[tex] \alpha |0\rangle + \beta |1 \rangle [/tex]
for some orthonormal basis [itex] |0\rangle, |1 \rangle [/itex]. The normalization conditions means we can associate this space with the unit 3-sphere
[tex] S^3 = \left\{ (z_1,z_2) \in \mathbb C^2 : |z_1|^2 + |z_2|^2 = 1 \right\} [/tex]
Now on the other hand, we also use the Bloch sphere to describe such states, where geometrically the Bloch sphere is just [itex] \mathbb{PC}^1[/itex] the complex projective line.
So I'm wondering, how do we map [itex] S^3 \to \mathbb{PC}^1 [/itex] quantum mechanically? What does this map look like? Is this occurring because we're throwing away the global phase in our transition to the Bloch sphere?
Something is telling me that Hopf fibrations are important here, but I just can't seem to put two and two together.