**5.2.1 Quantum transferrer** *π* → *o*<sup>2</sup>

Let us consider as initial state the polarization-encoded qubit

$$|\Psi\rangle\_{\rm in} = |\varrho\rangle\_{\pi}|0\rangle\_{\rm o} = (\mathfrak{a}|H\rangle\_{\pi} + \mathfrak{z}|V\rangle\_{\pi})|0\rangle\_{\rm o} \tag{35}$$

where |0�*<sup>o</sup>* indicates the TEM00 mode. By passing it through a pair of suitably oriented quarter-waveplates (one with the optical axis parallel to the horizontal direction and the other at 45◦), the photon state is rotated into the *L*, *R* basis:

$$(\mathfrak{a}|L\rangle\_{\pi} + \mathfrak{z}|R\rangle\_{\pi})|0\rangle\_{o} \tag{36}$$

After the QP the quantum state of the photon is then turned into the following:

$$
\langle \mathfrak{a} | \mathbb{R} \rangle | + \mathfrak{2} \rangle + \mathfrak{G} | L \rangle | - \mathfrak{2} \rangle. \tag{37}
$$

If a polarizer along the horizontal direction is used, we then obtain the state

$$|\Psi\rangle\_{\rm out} = |H\rangle\_{\pi} (\mathfrak{a}|+\mathfrak{2}\rangle\_{\mathfrak{o}\_2} + \mathfrak{z}|-\mathfrak{2}\rangle\_{\mathfrak{o}\_2}) = |H\rangle\_{\pi} |\mathfrak{g}\rangle\_{\mathfrak{o}\_2} \tag{38}$$

which completes the conversion. We note that such conversion process is probabilistic, since the state |Ψ�*out* is obtained with a probability *p* = 50%, owing to the final polarizing step. Moreover, since we are using the {|*H*�, |*V*�} basis for the polarization encoding and the *o*<sup>2</sup> = {| + 2�, | − 2�} for the OAM one, the transfer is associated also with a rotation of the Poincaré sphere.

20 Will-be-set-by-IN-TECH

Fig. 10. Experimental density matrices (real and imaginary parts) for the single photon entangled state (Nagali et al. (2009)). The computational values {0, 1} are associated to the {|*R*�, |*L*�} polarization states, and to {| + 2�, | − 2�} for the orbital angular momentum *m* for the first and the second qubit, respectively. The incoming state on the QP is (**a**) |*H*�*π*|0�*m*,and

two schemes, thus realizing the *bidirectional transfer* polarization-OAM-polarization ( *π* → *o*<sup>2</sup> → *π*). The latter demonstration is equivalent to demonstrate quantum communication using OAM for encoding the message. In other words, the qubit is initially prepared in the polarization space, then passed to OAM in a transmitting unit (Alice), sent to a receiving unit

where |0�*<sup>o</sup>* indicates the TEM00 mode. By passing it through a pair of suitably oriented quarter-waveplates (one with the optical axis parallel to the horizontal direction and the other

which completes the conversion. We note that such conversion process is probabilistic, since the state |Ψ�*out* is obtained with a probability *p* = 50%, owing to the final polarizing step. Moreover, since we are using the {|*H*�, |*V*�} basis for the polarization encoding and the *o*<sup>2</sup> = {| + 2�, | − 2�} for the OAM one, the transfer is associated also with a rotation of the Poincaré


(*α*|*L*�*<sup>π</sup>* + *β*|*R*�*π*)|0�*<sup>o</sup>* (36)

*α*|*R*�| + 2� + *β*|*L*�| − 2�. (37)


(Bob), where it is transferred back to polarization for further processing or detection.

After the QP the quantum state of the photon is then turned into the following:

If a polarizer along the horizontal direction is used, we then obtain the state

Let us consider as initial state the polarization-encoded qubit

at 45◦), the photon state is rotated into the *L*, *R* basis:

(**b**) |*V*�*π*|0�*m*.

sphere.

**5.2.1 Quantum transferrer** *π* → *o*<sup>2</sup>


Fig. 11. **Right Side -** Experimental density matrices *ρ* (the left column shows the real part and right column the imaginary part) measured for the output of the *π* → *o*<sup>2</sup> qubit transfer, for each of the three different predicted output states shown in the upper left corner of each row.**Left Side -** Experimental density matrices *ρ* (the left column shows the real part and right column the imaginary part) measured for the output of the *o*<sup>2</sup> → *π* qubit transfer, for each of the three different predicted output states shown in the upper left corner of each row. (Nagali et al. (2009))

The input arbitrary qubit is written in the polarization using two waveplates, as discussed previously. The experimental results for three specific choices of the input state are shown in Fig. (11). We find a good agreement with theory as demonstrated by the fidelity parameter, with an average fidelity value between the experimental states and the theoretical predictions equal to *F* = (97.7 ± 0.2)%.

Thus, we have demonstrated experimentally that the initial information encoded in an input TEM00 state can be coherently transferred to the OAM degree of freedom, thanks to the *π* → *o*<sup>2</sup> converter, giving rise to the preparation of a qubit in the orbital angular momentum. As the initial information has been stored in the orbital part of the qubit wave-function, new information can be stored in the polarization degree of freedom, allowing the transportation in a single photon of a higher amount, at least two qubits, of information.
