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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 tex2html_wrap_inline672 , i=0,1,2,3, arranged at the vertices of a regular tetrahedron. Choosing the same coupling structure tex2html_wrap_inline676 for all four polarizers the model leads to a homogeneous (in time) Poisson process on the sphere tex2html_wrap_inline678 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 tex2html_wrap_inline684 be the 2 by 2 matrices tex2html_wrap_inline690 , where tex2html_wrap_inline692 are the Pauli matrices, and let tex2html_wrap_inline694 be the four operators acting on tex2html_wrap_inline678 by tex2html_wrap_inline698 These operators play the role of Barnsley's affine transformations. To each transformation there is associated probability tex2html_wrap_inline700 , where tex2html_wrap_inline702 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 tex2html_wrap_inline478 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 tex2html_wrap_inline714 For details see [34].

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