Curriculum / Quantum Foundations / Measurement Bases
Measurement Bases
Understand computational and Hadamard bases and why choosing a basis matters.
Measurement Bases
Quantum mechanics doesn't have a single "correct" way to measure a qubit. The result you get depends on the basis you choose to measure in.
The Computational Basis (Z-basis)
The standard basis you've been using all along:
- •|0⟩ and |1⟩ are the eigenstates
- •
qc.measure()always measures in this basis - •A superposition (|0⟩ + |1⟩)/√2 collapses randomly to 0 or 1 with equal probability
The Hadamard Basis (X-basis)
Applying H before measuring effectively measures in the X-basis:
- •|+⟩ = (|0⟩ + |1⟩)/√2 and |−⟩ = (|0⟩ − |1⟩)/√2 are the eigenstates
- •A state in |0⟩ or |1⟩ collapses randomly in the X-basis
- •A state in |+⟩ collapses deterministically to 0 in the X-basis (after H)
Why Multiple Bases Matter
The power of quantum cryptography comes from this: measuring in the wrong basis destroys the original information. BB84 relies on Alice and Bob choosing bases randomly, and an eavesdropper who guesses wrong introduces detectable errors.
The Y-basis
Apply S† then H before measuring to access the Y-basis. The eigenstates are:
- •|i+⟩ = (|0⟩ + i|1⟩)/√2
- •|i−⟩ = (|0⟩ − i|1⟩)/√2
Basis Summary
| Basis | Eigenstates | How to measure |
|---|---|---|
| Z (computational) | |0⟩, |1⟩ | measure directly |
| X (Hadamard) | |+⟩, |−⟩ | apply H then measure |
| Y | |i+⟩, |i−⟩ | apply S† then H then measure |
Understanding these three bases gives you the full picture of single-qubit measurement.
How this lesson works
A guided reading lesson with interactive knowledge checks. Concepts are explained step by step with circuit diagrams and runnable examples, and you confirm understanding before moving on.
Part of: Quantum Foundations
Learn the basics: qubits, gates, superposition, and measurement.
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