# Entanglement, projection operator and partial trace

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1. Jan 9, 2016

### Matt atkinson

1. The problem statement, all variables and given/known data
Consider the following experiment: Alice and Bob each blindly draw a marble from a vase that contains one black and one white marble. Let’s call the state of the write marble $|0〉$ and the state of the black marble $|1〉$.
Consider what the state of Bob’s marble is when Alice finds a white marble
2. Relevant equations

3. The attempt at a solution
So I found the mixed state of Bob and alice's particle to be:
$$\rho=\frac{1}{2}|0,1\rangle \langle0,1|+\frac{1}{2}|1,0\rangle \langle1,0|$$
And i know that finding a white marble can be described in the following way:
$$\rho^B=\frac{Tr_A(|0\rangle_A\langle0|\rho)}{Tr(|0\rangle_A\langle0|\rho)}$$
where $Tr_A$ is the partial trace w.r.t Alice's system.
And just by reasoning i know the answer should be $|1\rangle\langle1|$ but im struggling to prove that by solving the above equation.
Here's my attempt:
$$\rho^B=\frac{Tr_A(|0\rangle_A\langle0|(|0,1\rangle \langle0,1|+|1,0\rangle \langle1,0|))}{Tr(|0\rangle_A\langle0|(|0,1\rangle \langle0,1|+|1,0\rangle \langle1,0|))}$$
$$\rho^B=\frac{Tr_A(|0\rangle_A\langle0|(|0\rangle \langle0| \otimes |1\rangle \langle1|+|1\rangle \langle1|\otimes|0\rangle \langle0|))}{Tr(|0\rangle_A\langle0|(|0\rangle \langle0| \otimes |1\rangle \langle1|+|1\rangle \langle1|\otimes|0\rangle \langle0|))}$$
But im not quite sure where to go from there, im a little inexperienced using Braket notation so any pointers would be greatly appreciated.

2. Jan 14, 2016