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I am reading

Exploring the quantum

By Serges Haroche & Jean-Michel Raimond

They consider a system $A$ living in $H_A$ surrounded by an environment $B$

Thus the problem lives in $H_A \otimes H_B$

On page 172, they want to explain the concept of generalized measurement.

They define it as operators living in $H_A$ that fullfils this condition :

$$ \sum_i M_i^{\dagger} M_i = 1 $$

And from what I understood, they say that during a measurement protocol, if the system $A$ is in a state $| \phi^A \rangle$ initially, and the environment in a state $|O^B\rangle$, we can see a measurement done on $A$ as :

  • First an entanglement that we write ($|u_i^B\rangle $ is orthogonal basis):

$$U_M (| \phi^A \rangle |O^B\rangle)=\sum_i (M_i | \phi^A \rangle ) \otimes |u_i^B\rangle $$

  • And then a projective measurement on the environment $B$ in the basis $|u_i^B\rangle $

And (again from what I understood), we will figure out that we find the same results with the $M_i$ as if it was a projective measurement. In practice we will end up with :

$\rho_A = Tr_B(\rho)=\sum_i M_i \rho M_i^{\dagger}$ if we don't read the measurement. And the measurement outcome $m_i$ associated to the $|u_i^B\rangle$ will have the probability $Tr(M_i \rho_A M_i^{\dagger})$

In practice, it is the result of the measurement of an observable in $H_B$ that will indirectly give us the state of the system in $H_A$.

My questions :

  • Did I understood the global picture ?
  • Why can we write a general entanglement process such as :

$$U_M (| \phi^A \rangle |O^B\rangle)=\sum_i (M_i | \phi^A \rangle ) \otimes |u_i^B\rangle $$

As the $M_i$ fullfils the condition $ \sum_i M_i^{\dagger} M_i = 1 $, for me, this unitary evolution is possibly not as general as possible ?

I don't get it.

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