Physics:Quantum Weak measurement: Difference between revisions
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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 | 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 | | author = Todd A. Brun | ||
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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. | 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. | ||
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== Concept == | == Concept == | ||
Revision as of 14:01, 17 May 2026
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.[1]
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.
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.[2]
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 interact with an ancilla in state . The joint state evolves under a weak interaction Hamiltonian
with unitary evolution
where 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 operators , with corresponding POVM elements
The post-measurement state conditioned on outcome is
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 ) controls this strength. In the limit:
- → projective (strong) measurement
- → 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.[3]
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.[4]
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)[5]
They are also closely related to the concept of the weak value, introduced by Aharonov, Albert, and Vaidman.[6]
See also
Table of contents (217 articles)
Index
Full contents
References
- ↑ Todd A. Brun (2002). "A simple model of quantum trajectories". Am. J. Phys. 70 (7): 719–737. doi:10.1119/1.1475328.
- ↑ Paul Busch (2009). No Information Without Disturbance: Quantum Limitations of Measurement. Springer.
- ↑ C. A. Fuchs; A. Peres (1996). "Quantum-state disturbance versus information gain". Phys. Rev. A 53: 2038–2045. doi:10.1103/PhysRevA.53.2038.
- ↑ A. Winter (1999). "Coding theorem and strong converse for quantum channels". IEEE Trans. Inf. Theory 45: 2481–2485. doi:10.1109/18.796385.
- ↑ S. J. Dolinar (1973). "An optimum receiver for the binary coherent state quantum channel". MIT Research Laboratory Report.
- ↑ Yakir Aharonov; David Z. Albert; Lev Vaidman (1988). "How the result of a measurement can be anomalous". Phys. Rev. Lett. 60: 1351–1354. doi:10.1103/PhysRevLett.60.1351.
Source attribution: Physics:Quantum Weak measurement
