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Weak measurement in quantum mechanics, a weak measurement is a type of quantum measurement that extracts only a small amount of information from a system while causing minimal disturbance to its state. Weak measurements arise naturally within the general framework of positive operator-valued measurements (POVMs), where the measurement strength can be continuously varied. In contrast to projective measurements, weak measurements only partially collapse the quantum state. A fundamental principle of quantum mechanics is that measurement disturbs the system being observed. According to results such as Busch’s theorem, there is no information gain without disturbance. Weak measurements operate in the regime where this disturbance is small. A standard description of weak measurement involves coupling the system weakly to an auxiliary system (ancilla).

Concept

A fundamental principle of quantum mechanics is that measurement disturbs the system being observed. According to results such as Busch’s theorem, there is no information gain without disturbance.^[1]

Weak measurements operate in the regime where this disturbance is small. As a result:

The measurement outcome carries limited information.
The post-measurement state remains close to the original state.
Repeated weak measurements can build up information gradually.

Weak interaction model

A standard description of weak measurement involves coupling the system weakly to an auxiliary system (ancilla).

Let a system in state $| ψ ⟩$ interact with an ancilla in state $| ϕ ⟩$ . The joint state evolves under a weak interaction Hamiltonian

$H = A \otimes B,$

with unitary evolution

$U \approx I - i λ A \otimes B - \frac{1}{2} λ^{2} A^{2} \otimes B^{2},$

where $λ$ is small. Because the evolution is close to the identity, the system is only weakly perturbed.

After measuring the ancilla, the system undergoes a transformation described by Kraus operatorss $M_{q}$ , with corresponding POVM elements

$E_{q} = M_{q}^{†} M_{q}, \sum_{q} E_{q} = I .$

The post-measurement state conditioned on outcome $q$ is

$| ψ_{q} ⟩ = \frac{M_{q} | ψ ⟩}{\sqrt{⟨ ψ | M_{q}^{†} M_{q} | ψ ⟩}} .$

This formalism shows that weak measurements are naturally embedded in the POVM framework.

Measurement strength

The strength of a measurement determines the tradeoff between information gained and disturbance caused:

**Strong measurement** → maximal information, large disturbance
**Weak measurement** → minimal disturbance, little information

In many models, a parameter (such as a Gaussian width $σ$ ) controls this strength. In the limit:

$σ \to 0$ → projective (strong) measurement
$σ \to \infty$ → very weak measurement

Information–disturbance tradeoff

Weak measurement illustrates the fundamental tradeoff between information gain and disturbance. This relationship has been studied extensively in quantum information theory.^[2]

A key result is the gentle measurement lemma, which states that if a measurement is unlikely to change the outcome, it only slightly disturbs the state.^[3]

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Physics:Quantum Weak measurement: Difference between revisions

Latest revision as of 11:32, 22 May 2026

Concept

Weak interaction model

Measurement strength

Information–disturbance tradeoff

Applications

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