Mathematics of Measurement then Partial Trace

Question

Say we have the following quantum state: $$ |\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle +|10\rangle)$$

To measure the first qubit and then further trace out the first qubit, my notes have the following equation for the post-measurement state:

$$ \rho = \frac{\text{Tr}_1(\Pi_1|\psi\rangle\langle\psi|)}{\text{Tr}(\Pi_1\otimes \mathbb{I} ​|\psi\rangle\langle\psi|)}$$

Firstly, if I wanted to measure the first qubit in the state $|0\rangle$, what would $\Pi_1$ be? Would it be the outer product $|0\rangle\langle0|$ because $|\psi\rangle$ is in outer product form?

Secondly, what would the mathematics look like to get the post-measurement state?

Answer 1

Yes, this is exactly what happens. Instead of saying "I want to measure this specific value," you specify a set of POVM elements asking "which of these values will I get?". When you are measuring the value of the first qubit alone to see if it is in state $|0\rangle$ or $|1\rangle$, the two POVM elements can be specified by $\Pi^{(0)}=|0\rangle\langle 0|\otimes \mathbb{I}$ and $\Pi^{(1)}=|1\rangle\langle 1|\otimes \mathbb{I}$. Then, if you measure the first qubit to be $|0\rangle$, meaning that your measurement device records some binary piece of information telling you that it has measured the first qubit to be $|0\rangle$, you follow the state-update rule provided with $\Pi^{(0)}$.

You should be able to do the rest from here! Just remember that the trace in the denominator can easily be manipulated into $$\mathrm{Tr}(\Pi^{(0)}|\psi\rangle\langle\psi|)=\langle\psi|\Pi^{(0)}|\psi\rangle=\langle\psi|(|0\rangle\langle 0|\otimes \mathbb{I})|\psi\rangle$$ but you cannot do the same with the partial trace in the numerator. For the partial trace, you need to remember the rule $$\mathrm{Tr}_1(X_{12})=(\langle 0|_1\otimes \mathbb{I}_2)X_{12}(| 0\rangle_1\otimes \mathbb{I}_2)+(\langle 1|_1\otimes \mathbb{I}_2)X_{12}(| 1\rangle_1\otimes \mathbb{I}_2).$$ In this last expression, I am explicitly using subscripts to indicate to which Hilbert space each item belongs - you can match all of the subscripts for each of the qubit states and the operators to do the calculations.