In quantum mechanics (and computation & 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.
[1] From Busch's theorem[2] any quantum system is necessarily disturbed by measurement, but the amount of disturbance is described by a parameter called the measurement strength.
[7] The most common methods of weak measurement are by coupling the quantum system to an ancilla qubit and projectively measuring the ancilla (which results in a weak measurement on the quantum system of interest), measuring a small part of large entangled systems, and for atomic physics, phase contrast imaging.
Consider using an ancilla, e.g. a field or a current, to probe a quantum system.
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).
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;[9][10] Viacheslav Belavkin;[11][12] Alberto Barchielli, L. Lanz, G. M. Prosperi;[13] Barchielli;[14] Carlton Caves;[15][16] Caves and Gerald J.
The notion of a weak measurement is often misattributed to Yakir Aharonov, David Albert and Lev Vaidman.
The Stern–Gerlach experiment is a quintessential example of the quantization of the electron spin angular momentum.
It involves a strong magnetic field gradient, which causes a spin-dependent force on electrons passing through the field, creating two pure-spin beams of electrons exiting the apparatus.
Suppose the magnet in this apparatus produced a very weak gradient, such as a sliver of calcite crystal.
There is no universally accepted definition of a weak measurement.
[20] The approach taken below is to interact two systems weakly and then measure one of them.
is the "interaction strength", which has units of inverse time.
gives Because it was only necessary to expand the unitary to a low order in perturbation theory, we call this is a weak interaction.
Further, the fact that the unitary is predominately the identity operator, as
The combined state of the system after interaction is Now we perform a measurement on the ancilla to find out about the system, this is known as an ancilla-coupled measurement.
Notice the ancilla system state records the outcome of the measurement.
With respect to the Kraus operators the post-measurement state of the combined system is The objects
Doing so gives the conditional state of the primary system alone: which we still label by the outcome of the measurement
We will use the canonical example of Gaussian Kraus operators given by Barchielli, Lanz, Prosperi;[13] and Caves and Milburn.
, where the position and momentum on both systems have the usual Canonical commutation relation
Take the initial wavefunction of the ancilla to have a Gaussian distribution The position wavefunction of the ancilla is The Kraus operators are (compared to the discussion above, we set
Phase-contrast imaging is an imaging method used in atomic physics, with cold and dense dilute gases of atoms, most commonly Bose–Einstein condensates.
As stated above, Busch's theorem[2] prevents a free lunch: there can be no information gain without disturbance.
[25] Recently the information-gain–disturbance tradeoff relation has been examined in the context of what is called the "gentle-measurement lemma".
[6][26] Since the early days it has been clear that the primary use of weak measurement would be for feedback control or adaptive measurements of quantum systems.
Indeed, this motivated much of Belavkin's work, and an explicit example was given by Caves and Milburn.
An early application of an adaptive weak measurements was that of Dolinar receiver,[27] which has been realized experimentally.
[20] Wiseman and Milburn's book[21] is a good reference for many of the modern developments.