In EEQT it is possible to model a simulataneous measurement of several non-commuting observables. And example would be a simultaneous measurement of the same component of position an momentum. This case, however, has not yet been studied - because of its computational difficulties. A simpler problem, namely that of a simultaneous measurement of several spin projections leads to chaotic behavior and fractal structure on the space of pure states. Following the discussion given in [33] let us couple a spin 1/2 quantum system to four yes-no polarizers corresponding to spin directions  , i=0,1,2,3, arranged at the vertices of a regular tetrahedron. Choosing the same coupling structure  for all four polarizers the model leads to a homogeneous (in time) Poisson process on the sphere  of norm 1 quantum spin states. The process is a non-linear version of Barnsley's iterated function system [32] and can be described as follows:
for i=0,1,2,3 let  be the 2 by 2 matrices  , where  are the Pauli matrices, and let  be the four operators acting on  by  These operators play the role of Barnsley's affine transformations. To each transformation there is associated probability  , where  is the radius-vector of the actual point on the spehere, that is to be transformed. Iteration leads to a self-similar structure, with sensitive dependence on the initial state and on the value of the coupling constant. Numerical simulation shows that when  decreases from 0.95 to 0.75, Hausdorff dimension of the limit set increases from 0.5 to 1.3. Fig. 1 shows a typical picture - here for  For details see [34].

Tue Dec 29 10:54:13 WET 1998