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Qubits & Quantum States

A qubit is the fundamental unit of quantum information. Unlike a classical bit which is definitively 0 or 1, a qubit exists in a superposition of both until measured. This topic establishes the mathematical framework — Dirac notation, the Bloch sphere, and the Born rule — that underpins all of quantum computing.

Key Concepts

Qubit
The quantum analogue of the classical bit. A qubit occupies a two-dimensional complex Hilbert space spanned by basis states ∣0⟩|0\rangle and ∣1⟩|1\rangle (the computational basis). Physically realized as electron spin, photon polarization, superconducting circuits, or trapped ion energy levels.
Superposition
A qubit can exist in any linear combination ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle where α,β∈C\alpha, \beta \in \mathbb{C}. The normalization constraint ∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1 ensures total probability equals one. Superposition is destroyed upon measurement.
Born Rule
When a qubit ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle is measured in the computational basis, the probability of obtaining ∣0⟩|0\rangle is ∣α∣2|\alpha|^2 and of obtaining ∣1⟩|1\rangle is ∣β∣2|\beta|^2. After measurement the qubit collapses to the observed state.
Bloch Sphere
A geometric representation of a single qubit state as a point on a unit sphere: ∣ψ⟩=cos⁡(θ/2)∣0⟩+eiϕsin⁡(θ/2)∣1⟩|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle. The north pole is ∣0⟩|0\rangle, south pole is ∣1⟩|1\rangle, and the equator holds equal superpositions. Global phase has no physical meaning.
Global Phase
Multiplying a state by eiγe^{i\gamma} (a global phase) produces no observable difference — all measurement probabilities are unchanged. Only relative phases between ∣0⟩|0\rangle and ∣1⟩|1\rangle components are physically meaningful.
Multi-Qubit States
An nn-qubit system lives in a 2n2^n-dimensional Hilbert space. Two qubits have basis states ∣00⟩,∣01⟩,∣10⟩,∣11⟩|00\rangle, |01\rangle, |10\rangle, |11\rangle. The exponential growth of Hilbert space dimension is the source of quantum computing's potential power.

Key Equations

General Qubit State
∣ψ⟩=α∣0⟩+β∣1⟩,α,β∈C|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad \alpha,\beta \in \mathbb{C}
Any single-qubit state is a complex linear combination of computational basis states.
Normalization
∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1
The sum of measurement probabilities must equal 1.
Born Rule
P(∣0⟩)=∣α∣2,P(∣1⟩)=∣β∣2P(|0\rangle) = |\alpha|^2, \quad P(|1\rangle) = |\beta|^2
Measurement probability equals the squared modulus of the amplitude.
Bloch Sphere Parametrization
∣ψ⟩=cos⁡ ⁣θ2∣0⟩+eiϕsin⁡ ⁣θ2∣1⟩|\psi\rangle = \cos\!\tfrac{\theta}{2}|0\rangle + e^{i\phi}\sin\!\tfrac{\theta}{2}|1\rangle
Every pure qubit state maps to a unique point on the unit sphere (θ∈[0,π], φ∈[0,2π)).
Inner Product
⟨ϕ∣ψ⟩=α∗γ+β∗δ for ∣ϕ⟩=α∣0⟩+β∣1⟩,  ∣ψ⟩=γ∣0⟩+δ∣1⟩\langle\phi|\psi\rangle = \alpha^*\gamma + \beta^*\delta \text{ for } |\phi\rangle=\alpha|0\rangle+\beta|1\rangle,\; |\psi\rangle=\gamma|0\rangle+\delta|1\rangle
The inner product measures overlap between quantum states.
Worked Example

Measurement Probabilities

Problem

A qubit is prepared in the state ∣ψ⟩=35∣0⟩+45∣1⟩|\psi\rangle = \frac{3}{5}|0\rangle + \frac{4}{5}|1\rangle. What are the probabilities of measuring ∣0⟩|0\rangle and ∣1⟩|1\rangle? Verify normalization.

Solution

Apply the Born rule to find each probability:

P(∣0⟩)=∣35∣2=925=0.36P(|0\rangle) = \left|\frac{3}{5}\right|^2 = \frac{9}{25} = 0.36
P(∣1⟩)=∣45∣2=1625=0.64P(|1\rangle) = \left|\frac{4}{5}\right|^2 = \frac{16}{25} = 0.64

Check normalization:

P(∣0⟩)+P(∣1⟩)=0.36+0.64=1.00✓P(|0\rangle) + P(|1\rangle) = 0.36 + 0.64 = 1.00 \checkmark
Answer P(|0⟩) = 0.36, P(|1⟩) = 0.64
Practice

Exercises

5 problems
1 of 5

For ∣ψ⟩=35∣0⟩+45∣1⟩|\psi\rangle = \frac{3}{5}|0\rangle + \frac{4}{5}|1\rangle, what is P(∣0⟩)P(|0\rangle)?

2 of 5

For ∣ψ⟩=12∣0⟩+12∣1⟩|\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle, what is P(∣1⟩)P(|1\rangle)?

3 of 5

A normalized qubit has α=0.6\alpha = 0.6 (real). What is ∣β∣2|\beta|^2?

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4 of 5

For ∣ψ⟩=i2∣0⟩+32∣1⟩|\psi\rangle = \frac{i}{2}|0\rangle + \frac{\sqrt{3}}{2}|1\rangle, what is P(∣0⟩)P(|0\rangle)?

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5 of 5

For ∣ψ⟩=12∣00⟩+12∣11⟩|\psi\rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle, what is the probability of measuring the first qubit as ∣0⟩|0\rangle?

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Key Takeaways

  • A qubit state ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle requires ∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1.
  • The Born rule: measurement yields ∣0⟩|0\rangle with probability ∣α∣2|\alpha|^2 and ∣1⟩|1\rangle with probability ∣β∣2|\beta|^2.
  • The Bloch sphere gives a complete geometric picture of all single-qubit pure states.
  • Global phase eiγe^{i\gamma} has no physical consequence; only relative phase matters.
  • An nn-qubit system lives in a 2n2^n-dimensional Hilbert space — the exponential foundation of quantum speedups.