Weak Quantum Measurement: Trajectory vs. Collapse

Qubit Preparation

Measurement Strength

Weak ↔ Strong 30%

Live Readout

Steps taken

Confidence

50%

Bloch z

0.00

Coherence |ρ01|

0.50

Information gained / step –

Disturbance caused / step –

A weak measurement extracts a little information and disturbs the state a little; a strong measurement extracts a lot — and disturbs a lot. You can't get one without the other.

How it works

Each tick, an unravelled quantum trajectory is computed by weighting the qubit's density matrix with a Gaussian measurement operator centred on ±1 along z. A noisy readout r is drawn, the state is Bayesian-updated toward whichever pole the readout favours, and the off-diagonal coherence is damped by the same operator — that damping is the measurement backaction.

P(r) = p₀·N(r; +1, σ²) + p₁·N(r; −1, σ²)
ρᵢᵢ' ∝ ρᵢᵢ · exp(−ε(r∓1)²)
ρ₀₁' ∝ ρ₀₁ · exp(−ε/2·[(r−1)²+(r+1)²])

Weak (small ε) — wide σ, a single readout is almost pure noise, the state barely moves. Repeat it thousands of times and the small nudges accumulate into a slow, visibly noisy walk toward a pole: a genuine continuous quantum trajectory.
Strong (large ε) — narrow σ, one readout already tells you almost everything, and the same step collapses coherence almost to zero: a single-shot projective measurement.
Information is estimated from how distinguishable the two Gaussian outcome distributions are at the current ε (a discrimination-probability measure); disturbance is the expected fractional loss of |ρ01| per step, integrated numerically at the current ε. They rise together — that coupling is the point.

COLLAPSED

Bloch vector Trajectory trail Strong collapse

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quantum trajectory — Bloch ⟨z⟩ over timeweak monitoring: noisy diffusion · strong: instant jump