Quantum Probability Tease (Single Qubit)
This page gives a friendly “probability-first” preview of how a single qubit produces outcomes that look
probabilistic, even though the state evolves deterministically under gates. The goal is intuition:
amplitudes \(\rightarrow\) squared magnitudes \(\rightarrow\) measurement probabilities.
1) What is a qubit?
A classical bit is either 0 or 1. A qubit can be in a superposition of both basis states.
We write the computational basis as \(\lvert 0\rangle\) and \(\lvert 1\rangle\), and a general pure qubit as:
\[
\lvert \psi\rangle = \alpha\lvert 0\rangle + \beta\lvert 1\rangle,
\qquad \alpha,\beta\in\mathbb{C},\qquad |\alpha|^2 + |\beta|^2 = 1.
\]
The complex numbers \(\alpha\) and \(\beta\) are called probability amplitudes. They are not probabilities
themselves — the probabilities come from their squared magnitudes.
2) Measurement probabilities (“amplitude squared” rule)
If you measure the qubit in the computational basis, the outcomes are probabilistic:
\[
P(\lvert 0\rangle) = |\alpha|^2,\qquad
P(\lvert 1\rangle) = |\beta|^2.
\]
Example: If \(\alpha=\beta=\tfrac{1}{\sqrt{2}}\), then \(P(\lvert 0\rangle)=P(\lvert 1\rangle)=\tfrac{1}{2}\).
That is exactly what happens when you apply a Hadamard gate to \(\lvert 0\rangle\).
3) Gates (unitary transformations)
In this preview, gates act like matrix multiplications on the state vector.
If we represent \(\lvert \psi\rangle\) as a column vector
\(\mathbf{v}=\begin{bmatrix}\alpha\\\beta\end{bmatrix}\),
then a gate \(U\) produces the new state \(\mathbf{v}'=U\mathbf{v}\).
The key point: gates change amplitudes, which changes probabilities — but in a structured way,
preserving normalization.
Common gates used here
\[
H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix},\quad
X=\begin{bmatrix}0&1\\1&0\end{bmatrix},\quad
Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}.
\]
Intuition:
- Hadamard \(H\) creates or removes superposition (often turning certainty into 50/50, depending on phase).
- Pauli-X \(X\) is a “bit flip”: it swaps \(\lvert 0\rangle\) and \(\lvert 1\rangle\).
- Pauli-Z \(Z\) is a “phase flip”: it changes the sign/phase of the \(\lvert 1\rangle\) amplitude.
4) Bloch sphere: a geometric picture
Any pure single-qubit state (up to a global phase) can be represented by a point on the unit sphere
called the Bloch sphere. A common parameterization is:
\[
\lvert \psi(\theta,\phi)\rangle
= \cos(\theta/2)\lvert 0\rangle + e^{i\phi}\sin(\theta/2)\lvert 1\rangle,
\qquad 0\le \theta \le \pi,\; 0\le \phi < 2\pi.
\]
From the state, the Bloch vector \((x,y,z)\) can be computed. For pure states, the \(z\)-coordinate determines
the computational-basis probabilities:
\[
P(\lvert 0\rangle)=\frac{1+z}{2},\qquad
P(\lvert 1\rangle)=\frac{1-z}{2}.
\]
In this tool, the animation shows the Bloch vector moving from the initial state to the final state after the gates.
The probability bars update during the animation using the pure-state relationship above.
5) Optional: rotation gates \(RX, RY, RZ\)
To connect to more advanced quantum notation, the tool also supports single-axis rotations:
\[
R_X(\theta)=\cos(\theta/2)I - i\sin(\theta/2)X,\quad
R_Y(\theta)=\cos(\theta/2)I - i\sin(\theta/2)Y,\quad
R_Z(\theta)=\begin{bmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{bmatrix}.
\]
These rotate the Bloch vector around the corresponding axis by angle \(\theta\).
6) What this preview is (and is not)
- Yes: single-qubit amplitudes, basic gates, and measurement probabilities in the computational basis.
- Yes: a geometric Bloch-sphere preview to build intuition.
- No: multi-qubit entanglement, tensor products, full measurement theory, or noise models.
If you want to extend this idea later, the next big step is two-qubit states and the appearance of entanglement,
where probabilities can no longer be explained by a single Bloch vector.
