Qubits & Superposition

The complex numbers $\alpha$ and $\beta$ are called amplitudes. They are not probabilities, they can be negative or complex, which is exactly what lets them interfere. The only constraint is normalization:

$|\alpha|^2 + |\beta|^2 = 1.$

A superposition is not mystical. It is a definite, fully specified vector, the state $|\psi\rangle$ is as concrete as a classical bit's value. What differs is how you get information out of it.

Measurement and the Born rule

Measuring a qubit in the computational basis returns one classical bit, and the outcome is random. The Born rule fixes the probabilities: you observe $0$ with probability $|\alpha|^2$ and $1$ with probability $|\beta|^2$. Normalization is what makes these sum to one.

Measurement also collapses the state. If you observe $0$, the qubit is now in state $|0\rangle$; measure again and you get $0$ with certainty. The pre-measurement amplitudes are gone, the act of reading is irreversible and destroys the superposition. This is the crux: from a single qubit you extract exactly one bit, and you never recover $\alpha$ and $\beta$ from one measurement. To even estimate them you must prepare the same state many times and gather statistics.

One more subtlety: a global phase is physically invisible. The states $|\psi\rangle$ and $e^{i\gamma}|\psi\rangle$ produce identical measurement statistics in every basis, so they are the same physical state. Only the relative phase between amplitudes, and their magnitudes, carry physical meaning.

The Bloch sphere

Two complex amplitudes are four real numbers. Normalization removes one degree of freedom and the irrelevance of global phase removes another, leaving two real parameters. Any pure single-qubit state can therefore be written

$|\psi\rangle = \cos\tfrac{\theta}{2}\,|0\rangle + e^{i\phi}\sin\tfrac{\theta}{2}\,|1\rangle, \qquad \theta \in [0, \pi],\ \phi \in [0, 2\pi),$

which is a point on the surface of a unit sphere, the Bloch sphere. The north pole is $|0\rangle$, the south pole is $|1\rangle$, and points on the equator are equal-weight superpositions distinguished only by the relative phase $\phi$. Antipodal points are orthogonal states. In the next lesson, quantum gates become rotations of this sphere. The picture is exact for one qubit and only one qubit: two qubits already live in a four-dimensional space with no such tidy visualization.

Worked example: the plus state and its cousin

Take the plus state

$|+\rangle = \tfrac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr),$

so $\alpha = \beta = \tfrac{1}{\sqrt{2}}$. The Born rule gives $|\alpha|^2 = |\beta|^2 = \tfrac{1}{2}$: measuring in the computational basis yields $0$ and $1$ with equal 50/50 odds. Now compare the minus state

$|-\rangle = \tfrac{1}{\sqrt{2}}\bigl(|0\rangle - |1\rangle\bigr).$

It also gives 50/50 outcomes in the computational basis, the squared magnitudes are identical. Yet $|+\rangle$ and $|-\rangle$ are different states, distinguished by the relative phase (the minus sign). That phase is invisible to a computational-basis measurement but becomes fully visible if you first rotate into the $X$ basis (apply a Hadamard, next lesson) and then measure: one gives $0$ with certainty, the other gives $1$. On the Bloch sphere the two sit at opposite ends of the equator ($+x$ and $-x$). This is the sharpest lesson of the whole topic: identical measurement probabilities do not imply identical states, phase is real physics that classical probability has no room for.

1. 11. PrepareCreate the plus state, an equal-weight superposition of the two basis states with a positive relative phase.
2. 22. Measure onceYou get a single bit, 0 or 1, each with probability one half. One run tells you almost nothing about the amplitudes.
3. 33. Repeat many times (shots)Over about a thousand runs the counts approach 500 and 500, estimating the squared magnitudes but not the phase.
4. 44. Notice the blind spotThese counts alone cannot tell the plus state from the minus state; the distinguishing phase only shows up after rotating the measurement basis.

Debunking "infinite parallelism"

The most repeated claim about quantum computing is that $n$ qubits "explore all $2^n$ possibilities at once," giving free exponential parallelism. It is false, and understanding why is essential.

What is true: the state of $n$ qubits is described by $2^n$ complex amplitudes, one per bitstring, so the description grows exponentially. What is false is that you can use them all. A single measurement of $n$ qubits returns $n$ classical bits, one bitstring, sampled according to the Born rule. You do not receive the $2^n$ amplitudes; you receive one sample and collapse the rest. There is no operation that reads out the exponential table hiding inside the state.

Real quantum speedups do not come from evaluating everything simultaneously and reading off the answer. They come from interference: an algorithm is engineered so that amplitudes for wrong answers cancel and amplitudes for right answers reinforce before you measure. That is a delicate, problem-specific design, and it is proven only for particular problems (Shor's factoring, Grover's quadratic search speedup), not as a blanket "everything at once."

Explaining superposition three ways

Further reading

• Nielsen & Chuang, Quantum Computation and Quantum Information, the standard graduate text; Chapters 1–2 cover qubits, the Born rule, and the Bloch sphere.
• IBM Quantum Learning, the successor to the Qiskit textbook, with runnable notebooks on states and measurement.
• John Preskill, Caltech Ph219 lecture notes, rigorous notes on quantum information from a leading researcher.
• Bloch sphere (Wikipedia), a concise reference for the geometric parametrization used above.