# 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 initial state in eigenstates of  , given by   and   (eigenvalues ). The time evolution of the 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 .

weak measurement procedure steps (i) and (ii)

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 .

Effect of post-selection

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 final state , which is called post-selection. In the 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 single Gaussian centered at , where is called the weak valueThe 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

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.,  .