# X basis Measurement in a mixed state with pure states

$$\newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|} %$$Suppose that we have a density operator $$\rho=\rho_1 \otimes\rho_2\otimes \rho_3$$ where $$\rho_1 ,\rho_3$$ are pure states and $$\rho_2$$ is a mixed state. If we perform a measurement on the whole system $$\rho$$ the state will remain mixed state. I am wondering how the probability outcomes of the state after the measurement can be computed.

For example suppose that the $$\rho_1,\rho_3$$ is a general state of a two qubit system where $$\rho=\ket{\psi}\bra{\psi}\,\,\text{with} \,\,\ket{\psi}=\cos(\theta/2)\ket{0}+e^{i\phi}\sin(\theta/2)\ket{1}$$ and the $$\rho_2=(1-p)\ket{\psi}\bra{\psi}+p I$$ where I is the identity operator. How can I compute the probabilities of the different outcomes of the density matrix due to measurement?

I am considering a projective measurement P where Pa =$$\ket{+}\bra{+}\otimes I\otimes I$$ or Pb= $$I\otimes \ket{+}\bra{+}\otimes I$$.

Is there a straightforward way to calculate the probability of the outcome in a statistical ensemble after the X basis measurement? Some posts claim that the answer uses the classical probability theory (Mixed state after measurement ).

• There isn't enough information here (specifically, regarding what measurement you want to describe) for this question to be really answerable. Aug 30, 2021 at 10:23
• Do you think it is still sloppy or undefined? If so I will erase it in order to write a specific example. Aug 30, 2021 at 10:34
• Greetings and welcome! Note that the culture in our community is that it's normal and expected for a question to be edited and improved after it's posted. However it's generally frowned upon for a user to delete a question and soon afterwards ask a very similar question — we have problems with people using that technique to avoid our question-review system. In its current form (v5) I don't see any reason for you to hide this question, but it's also fine if you improve it and use the "undelete" button later on.
– rob
Aug 30, 2021 at 12:38
• Thanks a lot for the answer. I didn't know if the question is fair enought to be answered. That's why I deleted it. If anyone can help with any literature or example I would be glad. Also if anybody wants more clarification about the problem please comment. Aug 30, 2021 at 12:59