Life & fun · November 17, 2024
Understanding the Bloch Sphere
Understanding the Bloch Sphere launched on November 17, 2024 with 7 votes and more than 75% of that year's launches. The site was still online in September 2026, 1 year on.
What it does
In other words, the state space of a single qubit is the set of unit vectors in the two dimensional complex vector space $\mathbb{C}^2$ with basis vectors $|0\rangle$ and $|1\rangle$. Operations on a single qubit are linear transformations which preserve the norm and therefore correspond to $2\times 2$ unitary matrices $U \in U(2)$. The vector space $\mathbb{C}^2$ is a four dimensional real vector space, and the constraint $|a|^2 + |b|^2 = 1$ means that the state space of a qubit can be identified with the three dimensional sphere in four dimensional space $S^3 \subset \mathbb{R}^4$. Concretely, if $a = x + yi$ and $b=z + wi$ then $|a|^2=x^2+y^2$ and $|b|^2 = z^2 + w^2$ and so the state…from daniellowengrub.com
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