Find four different complex numbers that all have exactly the same probability.
Hint: try 1, −1, i, −i Notice: |1|² = |−1|² = |i|² = |−i|² = 1 → Probability depends only on distance from origin, not direction.
what to discover
Drag a and b to move the point. Try all four quadrants. Probability depends only on distance from the origin — not on direction. A complex number is simply a pair of real numbers with a geometric representation.
Magnitude & Phase
areal part
1.000
bimaginary part
0.000
Cartesian → Polar
α = 1 + 0·i
↓
α = 1·ei·0°
r = √(a²+b²) = 1.000
φ = atan2(b, a) = 0°
Probability
P = |α|² = r² = 1.000
r²100%
physical meaning
Only the magnitude squared, |α|², gives the probability of measuring the corresponding basis state. The phase φ determines the direction of the complex vector — it does not change |α|², and therefore does not change the probability. Phase becomes important when multiple quantum amplitudes combine.
cartesian = polar
Both forms describe exactly the same complex number. r·eiφ separates magnitude (determines probability) from phase (determines direction of the vector).
α (|0⟩)
β (|1⟩)
Δφ (relative phase)
φα
0°
φβ
0°
Δφ ★
0°
α — amplitude of |0⟩
aα
1.000
bα
0.000
rα = 1.000 |α|² = 1.000
β — amplitude of |1⟩
aβ
0.000
bβ
0.000
rβ = 0.000 |β|² = 0.000
Quantum State
|ψ⟩ = 1|0⟩ + 0|1⟩
|ψ⟩ = α|0⟩ + β|1⟩
Global Phase γ
γrotates both together
0°
Both vectors rotate → Δφ stays fixed → qubit unchanged