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Convention: $|qubit_{A}, qubit_{B}\rangle$

The general Bell state equation: $|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k\oplus b\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\rangle + (-1)^{a}|1,1 \oplus b\rangle]$.

In mixing the states and expressing them as a matrix:

$\rho = \sum_{a,b = 0}^{1}p_{a,b}|\beta(a,b)\rangle \langle\beta(a,b)|$

where $p_{a,b} \geq 0$ and $\sum_{a,b=0}^{1}p_{a,b} = 1$.

How do I determine when $\rho$ is entangled?

I do know that entanglement is checked using either the Schmidt number (for state vector) or the partial transposition (density matrix). Since $\rho$ is a matrix, we can use take the partial transposition and apply it to the qubit B so that we have $\rho \rightarrow \rho^{T_{B}}$ More explicitly: $\rho = \sum_{a,b,a,b'}p_{a,b,a',b'}|A,B\rangle \langle A',B'| \rightarrow \rho^{T_{B}} = \sum_{a,b,a,b'}p_{a,b,a',b'}|A,B'\rangle\langle A,B|$

Any help is greatly appreciated.

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  • $\begingroup$ Didn't we have pretty much the same question last week? $\endgroup$
    – DaftWullie
    Oct 31, 2023 at 8:45
  • $\begingroup$ Yes but this involves entanglement condition. I left it out my mind last week to think about it but I can't seem to get past it. $\endgroup$
    – Physkid
    Oct 31, 2023 at 8:46
  • $\begingroup$ I hadn'y specifically meant one of your questions. I think it was this one... quantumcomputing.stackexchange.com/questions/34595/… $\endgroup$
    – DaftWullie
    Oct 31, 2023 at 12:51
  • $\begingroup$ @DaftWullie This is wild! The question looks similar. I wonder if it's a fellow classmate. I was referring to a question earlier I made on Bell state entanglement and presume you were referring to that. $\endgroup$
    – Physkid
    Nov 1, 2023 at 1:47

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