# Theoretical Framework

November 26, 2016 9:39 am“*How the Resulst of a Measurement of a Component of a Spin- Particle Can Turn Out to be 100″,*^{1} this is the rather *provocative* title of a paper published by Yakir Aharonov, David Z. Albert and Lev Vaidman (AAV) in early 1988. In this paper AAV propose what they call a “*new kind of quantum variable*” which they gave the unique name * weak value*.

Suppose an *object system* *S, *on which a Hermitian operator shall be measured and a *probe system P (often refereed to as ancilla and sometimes measurement device).* An important point is that * *both are considered being *quantum systems, *as illustrated below left*. *The weak measurement* *procedure involving three steps: *(i)* preparation of an initial quantum state of the object system *S*, this is called * pre-selection*. (ii) a weak coupling of this system with a

*probe system P*, with initial

*pointer state*, via a coupling Hamiltonian . The initial state of the probe system

*P*is supposed to be of Gaussian type (with initial spread ), having a

*q-*and a

*p-*representation given by and , respectively. The observables is the canonical variable of the probe system, with conjugate momentum and is a function with compact support near the time of the measurement – normalized such that its time integral is unity. The interaction is supposed to be sufficiently

*weak*, so that the system

*S*is only minimally disturbed. Using the spectral decomposition of the

*in eigenstates of , given by and (eigenvalues ). The time evolution of the*

**initial state***composite quantum*system, consisting of both the

*object*and the

*probe*system is given by . Evaluating this expression yields an

*entanglement*of probe and object system expressed as , which is a superposition a Gaussians (with width ) peaked at the

*eigenvalues*.

The probability distribution of the pointer states is given by . For this expression describes a usual *strong* measurement having the following three properties: i) the only possible measurement results are eigenvalues , ii) the probability of the outcome is given by the squared amplitude , iii) if the measurement yields system left in the eigenstate .

But what happens when the initial state of the probe system is broad ? In that case, instead of many peaks, we end up with a **single*** peak* , centered at the usual

*expectation value*of given by . If a

*strong measurement*of an observable is performed in addition, the system in put in an definite

*, which is called*

**final state***. In the*

**post-selection***q*-representation we get the following for the

*finial probe*state: , which is in

*p*-representation. At this point AAV introduce an approximation for this sum of Gaussians that consist only of a

*Gaussian centered at*

*single**,*where

*is called the*

*The important condition is that the width of the initial probe state is broad compared to the mean difference of the eigenstates of , denoted as . Using this approximation the final probe states is calculated as*

**weak value**.or .

At this point we want to take a look at a very simple example of the weak measurement, where a spin component of a spin- particle is weakly measured:

In a next step we calculate the output probability distribution of the final probe state for different values of the spread of the initial probe state. The animation below illustrates, that the AAV-approximation breaks down only for rather *small *values of the initial spread ,i.e., .

1. Yakir Aharonov, David Z. Albert, and Lev Vaidman, *Phys. Rev. Lett.* **60**, 1351–1354 (1988). ↩