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Sandbox Physics

Quantum Lab 01 · amplitude and phase

Quantum State & Measurement

Two Bloch-sphere angles control a pure quantum state. The vector shows its direction while the readings compare the Z basis with a rotated measurement basis. Change phase, notice what stays fixed, then rotate the basis to reveal interference.

Interactive modelQuantum State & Measurement
P(0)P(0) · Z basis50.0%
P(1)P(1) · Z basis50.0%
P(+β)P(+_\beta)85.4%
Relative phase45°

Physics tutorial

Qubits: a state is a probability structure, not an answer

BackgroundA classical bit is 0 or 1. A qubit uses two complex amplitudes and lets different measurement bases interrogate the same state.

Why it mattersPhase and measurement basis power quantum interference and are where the misleading phrase “both 0 and 1” causes the most confusion.

Start with the essentials

Focus question
If Z-basis probabilities stay 50/50, what physical meaning can changing relative phase have?
One-sentence intuition
Phase may leave probabilities in one basis unchanged while changing overlap with every other basis.

Core mathematical model

Bloch-sphere parameterization

∣ψ⟩=cos⁡θ2∣0⟩+eiϕsin⁡θ2∣1⟩|\psi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle

θ\theta controls basis-state weights and ϕ\phi is a relative phase readable through interference.

Z-basis Born probabilities

P(0)=cos⁡2θ2,P(1)=sin⁡2θ2P(0)=\cos^2\frac{\theta}{2},\qquad P(1)=\sin^2\frac{\theta}{2}

A measurement probability is the squared magnitude of its amplitude.

Common difficulties

Superposition is not a hidden classical answer

Typical misconceptionBefore measurement the qubit is already 0 or 1 and we simply do not know which.

Better mental modelOne state must predict multiple incompatible bases; in general there is no preassigned set of classical answers.

Global and relative phase differ

Typical misconceptionEvery complex phase changes experimental results.

Better mental modelMultiplying the whole state by eiγe^{i\gamma} changes no probability; only relative phase between components is observable.

Run the experiment

  1. 01

    Change amplitude weights

    Hold ϕ\phi fixed and sweep θ\theta from 0 to 180 degrees.

    What to observe: P(0)P(0) falls from 1 to 0 while the Bloch vector follows a meridian.
  2. 02

    Change phase only

    Set θ=90∘\theta=90^\circ and sweep ϕ\phi.

    What to observe: Z stays 50/50 while azimuth and other-basis probabilities change.
  3. 03

    Rotate the measurement basis

    Hold the state fixed and vary β\beta.

    What to observe: The same state gives different distributions in different bases.