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To perform Quantum Teleportation, we need state of two entangle qubits in the Bell basis $$\left(\frac{|00\rangle + \rangle11\rangle}{\sqrt{2}}, \frac{|00\rangle - |11\rangle}{\sqrt{2}}, \frac{|01\rangle + |10\rangle}{\sqrt{2}},\frac{|01\rangle - |10\rangle}{\sqrt{2}} \right).$$

To project in the product basis ($|00\rangle,|10\rangle,|01\rangle,|11\rangle$, or in the product basis along x or y axis (or any other axis)) we can check whether the individual spins are pointing up or down, but I don't know how the state is experimentally projected over a basis of entangled states like the Bell basis.

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Quantum circuit setup

Since the quantum circuits are all unitary operations, they are reversible too. The circuit that is used to generate Bell basis from the product basis can be used in reverse, just as shown below, to measure in the product basis, for which we already have an understanding.

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\begin{align*} |\Phi^+\rangle &=\frac1{\sqrt2}(|00\rangle+|11\rangle)\rightarrow|00\rangle \\ |\Psi^+\rangle &=\frac1{\sqrt2}(|01\rangle+|10\rangle)\rightarrow|01\rangle \\ |\Phi^-\rangle &=\frac1{\sqrt2}(|00\rangle-|11\rangle)\rightarrow|10\rangle \\ |\Psi^-\rangle &=\frac1{\sqrt2}(|01\rangle-|10\rangle)\rightarrow|11\rangle \end{align*}

Experimental implemetations

A photonic Bell state measurement setup consists of a beam-splitter followed by detectors. The two outputs of the beam-splitter each include a polarization beam-splitter followed by two single photon detectors and hence allow identifying two orthogonal polarization modes.

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                  Figure taken from R. Valivarthi et. al., Optics express, 22(20):24497–24506, 2014.

The input and output spatial modes are labelled 1,2 and 3,4, respectively. The same output port of the beam-splitter signals projection onto $|\psi^+\rangle$ and a coincidence of photons in modes 0 and 1 in different output ports of the beam-splitter is a signature of $|\psi^-\rangle$. However, the above setup cannot distinguish between states $|\phi+\rangle$ and $|\phi^-\rangle$. So with the above set-up we can unambiguously distinguish only 2 of the 4 Bell states.

A conceptually simple way to improve the efficiency of the BSM to 100% is to implement a CNOT (short for controlled not) gate using non-linear optics. A proposal for such an implementation to completely distinguish all four Bell states can be seen in this paper by Kim et. al. It must also noted that CNOT gate is not efficeintly acted using photons, but often implemented in atomic or stationary qubit systems.

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First, you apply a unitary transformation (that is, an interaction) which transforms the Bell states to a product basis. Then, you measure in that product basis.

For instance, a CNOT gate will transform the Bell state to the four $x$ axis eigenstates, or a controlled-phase (generated by a $\sigma_z\otimes \sigma_z$ interaction + local $z$ fields) to the $x$ and $z$ basis on the first/second qubit.

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