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3.2. Von Neumann formalism for quantum observables

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

${\textstyle Pr\{A=x||\rho \}=Tr[{\widehat {E}}^{A}(x)\rho ]=Tr[{\widehat {E}}^{A}(x)\rho {\widehat {E}}^{A}(x)].}$

$(5)$

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

${\textstyle \rho \rightarrow \rho _{x}={\frac {{\widehat {E}}^{A}(x)\rho {\widehat {E}}^{A}(x)}{Tr{\widehat {E}}^{A}(x)\rho {\widehat {E}}^{A}(x)}}}$

$(6)$

For reader’s convenience, we present these formulas for a pure initial state ${\textstyle \psi \in {\mathcal {H}}}$. The Born’s rule has the form:

${\textstyle Pr\{A=x||\rho \}=||{\widehat {E}}^{A}(x)\psi ||^{2}=<\psi \mid {\widehat {E}}^{A}(x)\psi >}$

$(7)$

The state transformation is given by the projection postulate:

${\textstyle \psi \rightarrow \psi _{x}={\widehat {E}}^{A}(x)\psi /\parallel {\widehat {E}}^{A}(x)\psi \parallel }$

$(8)$

Here the observable-operator ${\hat {A}}$ (its spectral decomposition) uniquely determines the feedback state transformations ${\textstyle {\mathcal {\Im }}_{A}(x)}$  for outcomes ${\textstyle x}$

${\textstyle \rho \rightarrow \Im _{A}(x)\rho ={\widehat {E}}^{A}(x)\rho {\widehat {E}}^{A}(x)}$

$(9)$

The map ${\textstyle \rho \rightarrow \Im _{A}(x)}$ given by (9) is the simplest (but very important) example of quantum instrument.