Quantum Information HT24, Bell basis
Flashcards
Define the elements of the Bell basis.
\[\begin{aligned}
\vert \Phi^+\rangle = \frac{ \vert 0\rangle \otimes \vert 0\rangle + \vert 1\rangle \otimes \vert 1\rangle}{\sqrt 2} \\\\
\vert \Phi^-\rangle = \frac{ \vert 0\rangle \otimes \vert 0\rangle - \vert 1\rangle \otimes \vert 1\rangle}{\sqrt 2} \\\\
\vert \Psi^+\rangle = \frac{ \vert 0\rangle \otimes \vert 1\rangle + \vert 1\rangle \otimes \vert 0\rangle}{\sqrt 2} \\\\
\vert \Psi^-\rangle = \frac{ \vert 0\rangle \otimes \vert 1\rangle - \vert 1\rangle \otimes \vert 0\rangle}{\sqrt 2}
\end{aligned}\]
Can you define the singlet state $ \vert \Psi^{-}\rangle$ and describe its special behaviour around measurements?
\[\vert \Psi^{-}\rangle = \frac{ \vert 0\rangle \otimes \vert 1\rangle - \vert 1\rangle \otimes \vert 0\rangle}{\sqrt 2}\]
If Alice and Bob perform measurements on the same basis, they will always obtain opposite outcomes, i.e.
\[p(0, 1) = p(1, 0) = \frac 1 2\] \[p(0, 0) = p(1, 1) = 0\]How can you relate the Bell basis $\{ \vert \Phi^+\rangle, \vert \Phi^-\rangle, \vert \Psi^+\rangle, \vert \Psi^-\rangle\}$ to the Pauli matrices, and state where this relationship is useful?
Let $U = [I, X, Y, Z]$. Then
\[\vert \Phi _ m\rangle = (U _ m \otimes I) \vert \Phi^+\rangle\]where we index the Bell basis as
\[\\{ \vert \Phi^+\rangle, \vert \Psi^+\rangle, \vert \Psi^-\rangle, \vert \Phi^-\rangle\\} = \\{ \vert \Phi _ 0\rangle, \vert \Phi _ 1\rangle, \vert \Phi _ 2\rangle, \vert \Phi _ 3\rangle\\}\]This is useful for quantum teleportation.