Quantum Information HT24, Bell basis
Flashcards
Define the elements of the Bell basis.
\[\begin{aligned}
|\Phi^+\rangle = \frac{|0\rangle \otimes |0\rangle + |1\rangle \otimes |1\rangle}{\sqrt 2} \\\\
|\Phi^-\rangle = \frac{|0\rangle \otimes |0\rangle - |1\rangle \otimes |1\rangle}{\sqrt 2} \\\\
|\Psi^+\rangle = \frac{|0\rangle \otimes |1\rangle + |1\rangle \otimes |0\rangle}{\sqrt 2} \\\\
|\Psi^-\rangle = \frac{|0\rangle \otimes |1\rangle - |1\rangle \otimes |0\rangle}{\sqrt 2}
\end{aligned}\]
Can you define the singlet state $ \vert \Psi^{-}\rangle$ and describe its special behaviour around measurements?
\[|\Psi^{-}\rangle = \frac{|0\rangle \otimes |1\rangle - |1\rangle \otimes |0\rangle}{\sqrt 2}\]
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
\[|\Phi_m\rangle = (U_m \otimes I) |\Phi^+\rangle\]where we index the Bell basis as
\[\\{|\Phi^+\rangle, |\Psi^+\rangle, |\Psi^-\rangle, |\Phi^-\rangle\\} = \\{|\Phi_0\rangle, |\Phi_1\rangle, |\Phi_2\rangle, |\Phi_3\rangle\\}\]This is useful for quantum teleportation.