# Find the Bell States of A and B in the following scenario

Let A has quantum memories $$M_1$$ and $$M_2$$ and B has quantum memories $$M_3$$ and $$M_4$$ capable of holding one qubit.

2 Bell states are shared among A & B in the following way:

First Bell state $$|B_{00}\rangle$$ has been shared among $$M_1$$ and $$M_3$$. Second Bell state $$|B_{00}\rangle$$ has been shared among $$M_2$$ and $$M_4$$.

Assume all calculations are in Bell basis.

How to calculate the Bell states of A[$$M_1$$ and $$M_2$$] and B[$$M_3$$ and $$M_4$$]?

• Why people are downvoting the question? Whats wrong in the question? Sep 28, 2019 at 14:45
• Please start using proper LaTeX formatting in your questions. Sep 29, 2019 at 9:33

Simply start by writing out everything $$|B_{00}\rangle_{13}|B_{00}\rangle_{24}=\frac12\left(|00\rangle_{13}|00\rangle_{24}+|00\rangle|11\rangle+|11\rangle|00\rangle+|11\rangle|11\rangle\right)$$ Let me rearrange each of these terms $$\frac12\left(|00\rangle_{12}|00\rangle_{34}+|01\rangle|01\rangle+|10\rangle|10\rangle+|11\rangle|11\rangle\right).$$ Now you can use the fact that the Bell states form a basis to rewrite each of those terms, although maybe it helps to notice that $$|00\rangle_{12}|00\rangle_{34}+|11\rangle|11\rangle=|B_{00}\rangle|B_{00}\rangle+|B_{01}\rangle|B_{01}\rangle$$ Hence, we get a final answer of $$\frac12\left(|B_{00}\rangle|B_{00}\rangle+|B_{01}\rangle|B_{01}\rangle+|B_{10}\rangle|B_{10}\rangle+|B_{11}\rangle|B_{11}\rangle\right)$$