Arjun Sahlot

Pure state · purity 1

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r = (0, 0, 1)|r| = 1

Amplitudes

(1)|0⟩ + (0)|1⟩

P(0), Z basis100%

P(1), Z basis0%

Density matrix & geometry

ρ =

1000

ρ = (I + xX + yY + zZ)/2. The diagonal gives Z probabilities; off-diagonal terms carry coherence.

A ket has two complex amplitudes. After normalization and ignoring global phase, a pure state needs two angles on a sphere in three real dimensions. Mixed states lie inside.

Purity Tr(ρ²) = (1 + |r|²)/2: 1 for pure, ½ at the center.

Apply a gate

Each click acts on the current state. Order matters.

Swap X and Z; reverse Y. H|0⟩ = |+⟩, H|+⟩ = |0⟩.

Rotate around

Angle90°

Gate identities to keep nearby

HH = XX = YY = ZZ = I. SS = Z; TT = S. HZH = X.

H exchanges X and Z information. It sends |0⟩ to |+⟩, but sends |+⟩ to |0⟩—the output is not always 50/50 in Z.

R_a(θ) = cos(θ/2)I − i sin(θ/2)σ_a. Unitary gates preserve norm and purity. A 360° rotation negates a ket; its Bloch point stays the same.

Apply a gate, then follow the state.

SequenceLeft to right · select a step to return

Opening qubit workspace…