Weak Measurement of a Qubit Superposition

Weak Continuous Measurement

Measurement strength κ 0.30

Rabi drive Ω 1.00

Qubit State

|ψ⟩ ≈ (|0⟩ + |1⟩)/√2

Coherence |⟨σx⟩,⟨σy⟩|1.00

P(|0⟩)

50.0%

P(|1⟩)

50.0%

|Bloch vector| r

1.00

Elapsed time

0.0

How it works

A qubit in superposition |ψ⟩ = α|0⟩ + β|1⟩ is a point on the Bloch sphere, r = (x, y, z), with the equator representing maximal superposition (equal probability of |0⟩ and |1⟩, full x–y coherence) and the poles representing the definite classical states |0⟩ and |1⟩.

Projective ("strong") measurement collapses the state to a pole instantly. Weak measurement instead couples the qubit only weakly and continuously to a detector, extracting a little information at a time while disturbing the state a little at a time. This is modelled by the stochastic master equation (homodyne unravelling of σ_z monitoring, Wiseman–Milburn / Jacobs form):

dx = -κ x dt + √κ · x z · dW
dy = -κ y dt + √κ · y z · dW
dz =  κ(1-z²) dt + √κ · (1-z²) · dW

where κ is the measurement rate and dW is Gaussian white noise (the random detector "clicks" of the readout). A Rabi drive Ω about the x-axis is added on top (dy += -Ω z dt, dz += Ω y dt), rotating the state and fighting the measurement's pull toward the poles.

κ = 0 — pure unitary Rabi rotation, the state precesses forever and never collapses.
κ large, Ω = 0 — rapid stochastic collapse to a random pole (the quantum Zeno-like limit of continuous strong readout).
Both nonzero — a tug-of-war: the drive keeps regenerating superposition while measurement keeps trying to destroy it, producing an irregular stochastic trajectory instead of a clean orbit.

The glowing trail is the actual quantum trajectory — one random realization of the measurement record, re-integrated every frame with a fresh random seed after Reset.

Bloch vector (state) Quantum trajectory

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