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State vectors and normalization
Represent |0⟩ as [1,0]ᵀ and |1⟩ as [0,1]ᵀ. A general pure state is [α,β]ᵀ with |α|²+|β|²=1. Global phase is unobservable, while relative phase affects interference.
The state vector stores amplitudes; measurement probabilities are squared magnitudes.
The Bloch sphere
A pure single-qubit state can be written cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩. The Bloch sphere is a geometric representation of θ and φ. Mixed states require a density matrix and lie inside the sphere.
The Bloch sphere is a map of single-qubit state space, not a literal physical sphere.
Measurement bases
Measurement need not be limited to the Z or computational basis. Rotating the qubit before measurement effectively measures in X or Y bases. Basis choice reveals different information about the state.
Quantum information is basis-dependent, which is central to tomography and algorithms.
Applied activity
Complete a simulator or analysis exercise: reproduce the lesson's central example, record assumptions and outputs, and explain one source of error or limitation.
Read the full lesson text
1. State vectors and normalization
Represent |0⟩ as [1,0]ᵀ and |1⟩ as [0,1]ᵀ. A general pure state is [α,β]ᵀ with |α|²+|β|²=1. Global phase is unobservable, while relative phase affects interference.
The state vector stores amplitudes; measurement probabilities are squared magnitudes.
2. The Bloch sphere
A pure single-qubit state can be written cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩. The Bloch sphere is a geometric representation of θ and φ. Mixed states require a density matrix and lie inside the sphere.
The Bloch sphere is a map of single-qubit state space, not a literal physical sphere.
3. Measurement bases
Measurement need not be limited to the Z or computational basis. Rotating the qubit before measurement effectively measures in X or Y bases. Basis choice reveals different information about the state.
Quantum information is basis-dependent, which is central to tomography and algorithms.
4. Applied activity
Complete a simulator or analysis exercise: reproduce the lesson's central example, record assumptions and outputs, and explain one source of error or limitation.
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