## 4 Quantum Integrated Photonic Devices
### Integrated Quantum Circuits for Quantum Cryptography
#### 4.2.2 Bell States Measurement Integrated Device for MDI  
Another interesting quantum passive device is a Bell states measurement device, which can be required in many optical quantum systems, e.g., in a MDI quantum cryptography system [217] in multicore optical fibers or few-mode optical fibers. This device can be implemented with integrated channel guides as shown next.  
In Figure 14 a sketch of this device is given for the case of two multicore optical fibers F\({}_{a}\) and F\({}_{b}\), where only two cores are shown. We must recall that multicore optical fibers and few-mode optical fibers allow for the increase of the optical transmission capability, therefore the use of these optical fibers for quantum communications is highly likely. Accordingly, the study of quantum operations with this kind of fiber also has a major  
Figure 13: An integrated device with 4 paths (single-mode channel guides) on a hybrid glass substrate is shown. An optical grating (OG) in the active glass section generates four laser weak signals obtained with pumping beams P\({}_{j}\), \((j=1,2,3,4)\); in the passive glass section, four directional couplers \(X_{\pi/4}\) and one coupler \(X_{\pi/2}\) along with a phase shifter \(Z_{\pi/2}\) generate the four states required for BB84 protocol at the outputs 2 and 3 coupled, for example, to two single-mode cores of a MCF.  
potential interest. The Bell projector can be implemented by a \(X_{\pi/2}\) coupler and two \(X_{\pi/4}\) couplers. Indeed, let us consider the following biphoton (or 2-qubit) Bell states, according to the notation used in Figure 14,  
\[\Psi^{\pm}=\frac{1}{\sqrt{2}}(|1_{a01}1_{b02}\rangle\pm|1_{a02}1_{b01}\rangle) \tag{12}\]  
By applying the optical transformations produced by \(X_{\pi/2}\) and \(X_{\pi/4}\) couplers for single-photon states given by Equations (5) and (6) one obtains  
\[\Psi^{+}\rightarrow\frac{i}{\sqrt{2}}(|1_{a1}1_{b1}\rangle+|1_{a2}1_{b2} \rangle),\quad\Psi^{-}\rightarrow\frac{1}{\sqrt{2}}(|1_{a1}1_{b2}\rangle+|1_{a 2}1_{b1}\rangle) \tag{13}\]  
Thus, when photons are registered in detectors \(\mathrm{D}_{a1},\mathrm{D}_{b1}\) or \(\mathrm{D}_{a2},\mathrm{D}_{b2}\) (photon coincidences) we have the Bell state \(\Psi^{+}\), and when photons are registered in detectors \(\mathrm{D}_{a1},\mathrm{D}_{b2}\) or \(\mathrm{D}_{a2},\mathrm{D}_{b1}\) we have the Bell state \(\Psi^{-}\). In short, an integrated Bell projector device has been obtained for Bell states \(\Psi^{\pm}\).  
On the other hand, this result has also an important impact in cryptography, in particular in a MDI quantum cryptography system. Indeed, let us consider the biphoton state \(|1_{a01}1_{b02}\rangle\) launched by [PERSON] and [PERSON], then the output state is given by  
\[|L\rangle=\frac{1}{2}\{(|1_{a1}1_{b2}\rangle-(|1_{a2}1_{b1}\rangle)+i(|1_{a1}1 _{b1}\rangle+|1_{a2}1_{b2}\rangle)\}\equiv\frac{1}{\sqrt{2}}(\Psi^{-}+i\Phi^{+}) \tag{14}\]  
that is, Bell states are again detected by photon concidences (Charlie device). Now, if the state launched by [PERSON] and [PERSON] is \(|1_{a02}1_{b01}\rangle\), the result is the same, except the sign: \((1/\sqrt{2})(-\Psi^{-}+i\Phi^{+})\); therefore, a quantum key distribution is possible because [PERSON] is not able to know which state has been sent, that is, \(|1_{a01}1_{b02}\rangle\) or \(|1_{a02}1_{b01}\rangle\).  
The above examples (quantum states generator and quantum projectors for BB84 and a Bell states measurement device) have made clear the potential of the integrated optical devices for the implementation of quantum operations. Next, we show how the practical implementation of these devices in a ion-exchanged glass platform is possible because only a few qubits (or qudits) are required.