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{{Quantum book backlink|Measurement and information}}

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
| 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.

[[File:Weak_measurement_diagram_yellow_bg.png|thumb|400px|right|Weak measurement gently probes a quantum state, extracting limited information while only slightly disturbing the system.]]

== 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.<ref name=Busch2009>{{cite book |author=Paul Busch |title=No Information Without Disturbance: Quantum Limitations of Measurement |publisher=Springer |year=2009}}</ref>

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 <math>|\psi\rangle</math> interact with an ancilla in state <math>|\phi\rangle</math>. The joint state evolves under a weak interaction Hamiltonian

<math display="block">
H = A \otimes B,
</math>

with unitary evolution

<math display="block">
U \approx I - i\lambda A \otimes B - \frac{1}{2}\lambda^2 A^2 \otimes B^2,
</math>

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

<math display="block">
E_q = M_q^\dagger M_q, \quad \sum_q E_q = I.
</math>

The post-measurement state conditioned on outcome <math>q</math> is

<math display="block">
|\psi_q\rangle = \frac{M_q |\psi\rangle}{\sqrt{\langle \psi | M_q^\dagger M_q | \psi \rangle}}.
</math>

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 <math>\sigma</math>) controls this strength. In the limit:

* <math>\sigma \to 0</math> → projective (strong) measurement
* <math>\sigma \to \infty</math> → 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.<ref name=FuchsPeres>{{cite journal
| author1=C. A. Fuchs
| author2=A. Peres
| title=Quantum-state disturbance versus information gain
| journal=Phys. Rev. A
| volume=53
| year=1996
| pages=2038–2045
| doi=10.1103/PhysRevA.53.2038
}}</ref>

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.<ref name=Winter1999>{{cite journal
| author=A. Winter
| title=Coding theorem and strong converse for quantum channels
| journal=IEEE Trans. Inf. Theory
| volume=45
| year=1999
| pages=2481–2485
| doi=10.1109/18.796385
}}</ref>

== Applications ==
Weak measurements are widely used in:

* [[Quantum control]] and feedback systems
* Continuous quantum measurement and quantum trajectories
* [[Quantum information]] processing
* Precision measurement and amplification techniques
* Adaptive measurement strategies (e.g. the Dolinar receiver)<ref name=Dolinar>{{cite journal
| author=S. J. Dolinar
| title=An optimum receiver for the binary coherent state quantum channel
| journal=MIT Research Laboratory Report
| year=1973
}}</ref>

They are also closely related to the concept of the [[weak value]], introduced by Aharonov, Albert, and Vaidman.<ref name=Aharonov1988>{{cite journal
| author1=Yakir Aharonov
| author2=David Z. Albert
| author3=Lev Vaidman
| title=How the result of a measurement can be anomalous
| journal=Phys. Rev. Lett.
| volume=60
| year=1988
| pages=1351–1354
| doi=10.1103/PhysRevLett.60.1351
}}</ref>

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Weak measurement|1}}

Physics:Quantum Weak measurement - Revision history

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