Revision as of 08:14, 20 May 2026

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Weak measurement is a Book I topic in the Quantum Collection. 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. 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.{{cite journal | title = A simple model of quantum trajectories Weak measurements arise naturally within the general framework of positive operator-valued measurements (POVMs), where the measurement strength can be continuously varied.

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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 {{Short description|Quantum Collection topic on Quantum Weak measurement}}
 {{Quantum book backlink|Measurement and information}}
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 <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-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.<ref name=Brun2002>{{cite journal
+'''Weak measurement''' is a Book I topic in the Quantum Collection. 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. 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.{{cite journal | title = A simple model of quantum trajectories Weak measurements arise naturally within the general framework of positive operator-valued measurements (POVMs), where the measurement strength can be continuously varied.
- | author = Todd A. Brun
- | title = A simple model of quantum trajectories
- | journal = Am. J. Phys.
- | volume = 70
- | issue = 7
- | pages = 719–737
- | year = 2002
- | doi = 10.1119/1.1475328
- }}</ref>
-Weak measurements arise naturally within the general framework of [[POVM|positive operator-valued measurements]] (POVMs), where the measurement strength can be continuously varied. In contrast to [[projective measurement]]s, weak measurements only partially collapse the quantum state.
 </div>
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 where <math>\lambda</math> 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 operator]]s <math>M_q</math>, with corresponding POVM elements
+After measuring the ancilla, the system undergoes a transformation described by Kraus operatorss <math>M_q</math>, with corresponding POVM elements
 <math display="block">
@@ Line 106: / Line 95: @@
 Weak measurements are widely used in:
-* [[Quantum control]] and feedback systems
+* Quantum control and feedback systems
 * Continuous quantum measurement and quantum trajectories
-* [[Quantum information]] processing
+* Quantum information processing
 * Precision measurement and amplification techniques
 * Adaptive measurement strategies (e.g. the Dolinar receiver)<ref name=Dolinar>{{cite journal

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Physics:Quantum Weak measurement: Difference between revisions

Revision as of 08:14, 20 May 2026

Concept

Weak interaction model

Measurement strength

Information–disturbance tradeoff

Applications

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