Weak measurement

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. 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.

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.

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.

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. 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.