Superposition

Learning Objectives

• Define superposition as a linear combination of basis states.
• Differentiate coherent superposition from classical uncertainty.
• Explain why relative phase controls interference.

Scene

In the next chamber, Alice sees two mirrored paths merge into one detector. The guide says: “Your probabilities are not enough. You also need phase.”

Conceptual explanation

A classical statement “the bit is either 0 or 1, and I do not know which” describes ignorance about a pre-existing value.
A quantum superposition describes a state itself:

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle.$$

This is not merely hidden classical information. Coherence and relative phase allow interference.

Hadamard examples

Equal superposition from |0⟩

H|0⟩ = (|0⟩ + |1⟩) / √2

Equal superposition with relative minus sign

H|1⟩ = (|0⟩ - |1⟩) / √2

The minus sign matters. Both states give 50/50 outcomes if measured immediately in the computational basis, but they behave differently under later interference operations.

Thought experiment: interference

Prepare either $|+\rangle$ or $|-\rangle$, then apply another Hadamard:

• $H|+\rangle = |0\rangle$
• $H|-\rangle = |1\rangle$

The relative minus sign changes the final deterministic result. This is impossible to explain using only “unknown classical bit” language.

Check Your Understanding

Why does the minus sign in (|0⟩ - |1⟩)/√2 matter?

It does not matter because probabilities are always 50/50.It changes relative phase, which can change outcomes after later gates.It means the state is not physically valid.

Check AnswerTry Again

Summary

Superposition is a coherent linear-combination model, not classical ignorance. Relative phase is central because amplitudes interfere under subsequent gates. Next we formalize measurement and the Born rule.