Bloch Coordinates (x, y, z)

Every qubit state can be described by a point on the Bloch sphere — a unit sphere whose poles are $|0\rangle$ (north) and $|1\rangle$ (south). The Cartesian coordinates $(x, y, z)$ of that point form the Bloch vector, and knowing how to compute them from the amplitudes is the key to reading the Bloch sphere display.

From amplitudes to the density matrix

A single-qubit pure state is written

where $\alpha, \beta \in \mathbb{C}$. Its density matrix is the outer product $\rho = |\psi\rangle\langle\psi|$:

The diagonal entries $\rho_{00} = |\alpha|^2$ and $\rho_{11} = |\beta|^2$ are the measurement probabilities. The off-diagonal entry $\rho_{01} = \alpha\overline{\beta}$ encodes the coherence — the extent to which the state can interfere.

The Bloch vector formulas

The Bloch coordinates are read directly from the density matrix:

These three formulas have clean geometric meaning:

• $z$ measures the bias between the two poles. If $|\alpha|^2 = 1$ the state is $|0\rangle$ and $z = 1$; if $|\beta|^2 = 1$ the state is $|1\rangle$ and $z = -1$; equal probabilities give $z = 0$ (equator).
• $x$ and $y$ capture the relative phase between $\alpha$ and $\beta$. When both amplitudes are real and positive ($\beta$ has phase $0$), $y = 0$ and $x > 0$, placing the state on the $+x$ axis — that is exactly $|+\rangle$.

Worked example

Consider the state with real amplitudes $\alpha = \tfrac{\sqrt{3}}{2}$ and $\beta = \tfrac{1}{2}$. Since both amplitudes are real, $\rho_{01} = \alpha\beta$ is real too and $\mathrm{Im}(\rho_{01}) = 0$, so $y = 0$.

A quick sanity check: $x^2 + y^2 + z^2 = \tfrac{3}{4} + 0 + \tfrac{1}{4} = 1$. ✓

This state sits in the $xz$-plane, tilted $60°$ from the north pole (since $z = \cos 60° = \tfrac{1}{2}$).

Try it

Apply the $z$ formula to the example above and return the z-coordinate. Math is available for Math.sqrt.

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