Notes on Bloch Sphere

The purpose of this post is to record my notes on the Bloch Sphere in the context of quantum information and quantum computing. 

- The Bloch Sphere is a 3D visual representation of the state of a qubit. You can think of it as the spin angular momentum vector for an electron (or any spin 1/2 particle for that matter which lives in a 2D complex Hilbert space). 

- In actuality, the Bloch Sphere is representing the quantum state of a qubit in a subset of its Hilbert space which is 2D with respect to complex numbers and isomorphically 4D with respect to real numbers. By placing quantum mechanical constraints on that full Hilbert space, we reach a space equivalent to the 2D unit sphere S^2 as described in later bullet points.

- The axes of the grid being used to visualize the qubit state vector are X, Y and Z corresponding to the different Pauli spin matrices or the different directions along which spin can be measured in the spin 1/2 particle case.

- Heuristically, it makes sense that you would need 3 dimensions to represent the state of a qubit since it really has 4 degrees of freedom (2 for each of its complex coefficients [c1, c2] corresponding to the 0 and 1 basis states in the Z basis for example), with one removed for 4-1 = 3 due to the normalization condition that the sum of the square norms of the basis coefficients must sum to 1.

- In fact, the Bloch Sphere itself is a 2D surface equivalent to S^2 (the 2D circle). We get here by first noting that the normalization condition requires Re(c1)^2 + Im(c1)^2 + Re(c2)^2 + Im(c2)^2 = 1 which yields S^3 as implied in the above bullet point. Next, we can form equivalence classes of the points on S^3 by grouping together states that differ by merely a global phase exp(i*phi). Topologically, this amounts to dividing S^3 by U(1) i.e. S^3/U(1) = S^3/S^1 = S^2. 

References:

- Check out this great interactive visualization to play with the Bloch Sphere. v0.12 Grokking the Bloch Sphere (javafxpert.github.io)