Physics:Quantum Weak measurement

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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.[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.

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

H=A⊗B,

with unitary evolution

U≈I−iλA⊗B−12λ2A2⊗B2,

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 operators Mq, with corresponding POVM elements

Eq=Mq†Mq,∑qEq=I.

The post-measurement state conditioned on outcome q is

|ψq⟩=Mq|ψ⟩⟨ψ|Mq†Mq|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→0 → projective (strong) measurement
  • σ→∞ → 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 (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ Todd A. Brun (2002). "A simple model of quantum trajectories". Am. J. Phys. 70 (7): 719–737. doi:10.1119/1.1475328. 
  2. ↑ Paul Busch (2009). No Information Without Disturbance: Quantum Limitations of Measurement. Springer. 
  3. ↑ C. A. Fuchs; A. Peres (1996). "Quantum-state disturbance versus information gain". Phys. Rev. A 53: 2038–2045. doi:10.1103/PhysRevA.53.2038. 
  4. ↑ A. Winter (1999). "Coding theorem and strong converse for quantum channels". IEEE Trans. Inf. Theory 45: 2481–2485. doi:10.1109/18.796385. 
  5. ↑ S. J. Dolinar (1973). "An optimum receiver for the binary coherent state quantum channel". MIT Research Laboratory Report. 
  6. ↑ 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. 
Author: Harold Foppele

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