Modifying a quantum teleportation protocol step.

Question

Assuming that Alice and Bob share an entangled state $\left|\beta_{00}\right\rangle_{(23)}=(|00\rangle+|11\rangle) / \sqrt{2}$. Also, Alice has an additional qubit in state $|\psi\rangle_{(1)} .$ As step for the quantum teleportation protocol Allice applies the unitary $H_{(1)} C(X)_{(12)}$ on her qubits and then performs the measurement $\{|a b\rangle\langle a b|: a, b \in\{0,1\}\}$. How can I find a projective measurement $\left\{P_{a b}\right\}$ such that we can replace change this step into only performing the measurement $\left\{P_{a b}: a, b \in\{0,1\}\right\}$ on Alice's qubits.

Answer 1

Mathematically, the answer is straightforward: project onto the Bell states. You will have to choose some labeling scheme for your projectors, but they will always be the set $$\{P_{ab}\}=\{|\Phi^+\rangle\langle\Phi^+|,|\Phi^-\rangle\langle\Phi^-|,|\Psi^+\rangle\langle\Psi^+|,|\Psi^-\rangle\langle\Psi^-|\},$$ where the four Bell states are defined as $$|\Phi^+\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}},\,|\Phi^-\rangle=\frac{|00\rangle-|11\rangle}{\sqrt{2}},\,|\Psi^+\rangle=\frac{|01\rangle+|10\rangle}{\sqrt{2}},\,|\Psi^-\rangle=\frac{|01\rangle-|10\rangle}{\sqrt{2}}.$$

Physically, this is a more intricate question. The most straightforward way to do this measurement, which involves projected onto entangled states, is to do the exact unitary that you mention (this unitary uses the CNOT gate, which is an entangling gate). A Bell state measurement can be done with some physical systems (ion traps) more easily than with others (photons).