# Generalized measurement and entanglement operation

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.