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===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 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math display="inline">Pr\{A=x||\rho\}=Tr[\widehat{E}^A(x)\rho]=Tr[\widehat{E}^A(x)\rho\widehat{E}^A(x)]</math>
| width="33%" align="right" |<math>(5)</math>
|}


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

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math display="inline">\rho\rightarrow\rho_x=\frac{\widehat{E}^A(x)\rho\widehat{E}^A(x)}{Tr\widehat{E}^A(x)\rho\widehat{E}^A(x)}
</math>
| width="33%" align="right" |<math>(6)</math>
|}


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: 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math display="inline">Pr\{A=x||\rho\}=||\widehat{E}^A(x)\psi||^2=<\psi\mid\widehat{E}^A(x)\psi></math>
| width="33%" align="right" |<math>(7)</math>
|}


The state transformation is given by the projection postulate: 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math display="inline">\psi\rightarrow\psi_x=\widehat{E}^A(x)\psi/\parallel\widehat{E}^A(x)\psi\parallel</math>
| width="33%" align="right" |<math>(8)</math>
|}


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
</math>

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math display="inline">\rho\rightarrow\Im_A(x)\rho=\widehat{E}^A(x)\rho\widehat{E}^A(x)</math>
| width="33%" align="right" |<math>(9)</math>
|}


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