In quantum mechanics (and computation and information), weak measurement is a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little. From Busch's theorem any quantum system is necessarily disturbed by measurement, but the amount of disturbance is described by a parameter called the measurement strength. Weak measurement is a subset of the more general form of quantum measurement described by operators known as POVMs, where the strength of measurement is low. In the literature weak measurements are also known as unsharp, fuzzy, dull, noisy, approximate, and gentle measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.
History Weak measurements were first thought about in the context of weak continuous measurements of quantum systems (i.e. quantum filtering and quantum trajectories). The physics of continuous quantum measurements is as follows. Consider using an ancilla, e.g. a field or a current, to probe a quantum system. The interaction between the system and the probe correlates the two systems. Typically the interaction only weakly correlates the system and ancilla (specifically, the interaction unitary operator need only to be expanded to first or second order in perturbation theory). By measuring the ancilla and then using quantum measurement theory, the state of the system conditioned on the results of the measurement can be determined. In order to obtain a strong measurement, many ancilla must be coupled and then measured. In the limit where there is a continuum of ancilla the measurement process becomes continuous in time. This process was described first by: Michael B. Mensky; Viacheslav Belavkin; Alberto Barchielli, L. Lanz, G. M. Prosperi; Barchielli; Carlton Caves; Caves and Gerard J. Milburn. Later on Howard Carmichael and Howard M. Wiseman also made important contributions to the field. The notion of a weak measurement is often misattributed to Yakir Aharonov, David Albert and Lev Vaidman. In their article they consider an example of a weak measurement (and perhaps coin the phrase "weak measurement") and use it to motivate their definition of a weak value, which they defined there for the first time.
Theory: Coupling to ancilla There is no universally accepted definition of a weak measurement. One approach is to declare a weak measurement to be a generalized measurement where some or all of the Kraus operators are close to the identity. The approach taken below is to interact two systems weakly and then measure one of them. After detailing this approach we will illustrate it with examples.
Weak interaction and ancilla-coupled measurement Consider a system that starts in the quantum state | ψ ⟩ {\displaystyle |\psi \rangle } and an ancilla that starts in the state | ϕ ⟩ {\displaystyle |\phi \rangle } . The combined initial state is | Ψ ⟩ = | ψ ⟩ ⊗ | ϕ ⟩ {\displaystyle |\Psi \rangle =|\psi \rangle \otimes |\phi \rangle } . These two systems interact via the Hamiltonian H = A ⊗ B {\displaystyle H=A\otimes B} , which generates the time evolutions U ( t ) = exp [ − i x t H ] {\displaystyle U(t)=\exp[-ixtH]} (in units where ℏ = 1 {\displaystyle \hbar =1} ), where x {\displaystyle x} is the "interaction strength", which has units of inverse time. Assume a fixed interaction time t = Δ t {\displaystyle t=\Delta t} and that λ = x Δ t {\displaystyle \lambda =x\Delta t} is small, such that λ 3 ≈ 0 {\displaystyle \lambda ^{3}\approx 0} . A series expansion of U {\displaystyle U} in λ {\displaystyle \lambda } gives
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