Difference between revisions of "Store:QLMen06"

===3.2. Von Neumann formalism for quantum observables===

In the original quantum formalism (Von Neumann, 1955), physical observable <math>A</math> is represented by a Hermitian operator <math>\hat{A}</math> . We consider only operators with discrete spectra:<math>\hat{A}=\sum_x x\hat{E}^A(x)</math> where <math>\hat{E}^A(x)</math> is the projector onto the subspace of <math display="inline">\mathcal{H}</math>  corresponding to the eigenvalue <math display="inline">x</math>. Suppose that system’s state is mathematically represented by a density operator<math display="inline">\rho</math>. Then the probability to get the answer <math display="inline">x</math> is given by the Born rule

and according to the projection postulate the post-measurement state is obtained via the state-transformation:  

 

For reader’s convenience, we present these formulas for a pure initial state <math display="inline">\psi\in\mathcal{H}</math>. The Born’s rule has the form:  

 

The state transformation is given by the projection postulate:  

 

Here the observable-operator <math>\hat{A}</math> (its spectral decomposition) uniquely determines the feedback state transformations  <math display="inline">\mathcal{\Im}_A(x)</math>  for outcomes <math display="inline">x

The map <math display="inline">\rho\rightarrow\Im_A(x)</math> given by (9) is the simplest (but very important) example of quantum instrument.