## 4 Quantum Integrated Photonic Devices
### Fundamental Aspects of Quantum Integrated Optical Components  
The use of quantum light states in integrated optical devices allows to develop different on-chip applications in quantum optical communications and in other fields such as quantum photonic simulations [196] and quantum photonic sensing [218]. Rigorous theories of linear and nonlinear quantum spatial propagation states in integrated waveguides can be used [219; 220; 221; 222], which allows us to study and design linear and nonlinear quantum integrated optical elements. Moreover, we consider three types of quantum states of a great interest for proofs of concept as the coherent states, the single photon states and the biphoton states. Single photon states, and in particular 1-ququart states (dimension \(d=4\)), have a high interest, for instance, in quantum cryptography; 1-ququart are the states giving rise to the beginning of the so called high dimension quantum cryptography with 1-qudits (\(d=N>2\)), that is,  
\[|L_{s}\rangle=\sum_{j=1}^{N}c_{j}|0\ldots 1_{j}\ldots 0\rangle\equiv\sum_{j=1} ^{N}c_{j}|1_{j}\rangle \tag{4}\]  
where subscript \(s\) stands for single-mode photon. This single photon state \(|L_{s}\rangle\) shows that the photon is in a quantum superposition in the \(N\) channel waveguides. Moreover, by using 1-qudits states we can form quantum bases of orthogonal states. These bases have to be in turn mutually unbiased bases (MUBs), which requires \(|c_{1}|=\ldots=|c_{N}|\)[198]. On the other hand, lasers emit coherent states \(|\alpha\rangle\), with \(\alpha\in\mathbb{C}\), then by using, for instance, a 1 xN splitter or a cascade of directional couplers [51] a multimode coherent state \(|a_{1}\ldots a_{N}\rangle\) is obtained. It is well known, however, that for weak coherent (wc) states regime, \(|a_{i}|\ll 1\), an approximated multimode single photon state is achieved, that is, \(|L_{wc}\rangle\approx|0\ldots 0\rangle+\sum_{j=1}^{N}\alpha_{j}|0\ldots 1_{j} \ldots 0\rangle\). These states are less efficient because there is a single photon state with a small probability, but it is enough for quantum cryptography. Next, it is worth describing the quantum mechanical treatment of the basic passive integrated optical elements of our integrated circuits, that is, the directional couplers \(2\times 2\) and phase shifters. We must stress that these \(2\times 2\) integrated optical elements (couplers and phase shifters) are enough to generate unitary transformations of dimension \(N\), that is, \(\mathrm{SU}(N)\) transformations, by concatening properly such optical elements [216]. Let us start by considering a two-dimensional quantum problem (\(N=2\)), that is, the single photon quantum states are 1-qubits, namely, \(|L_{s}\rangle=c_{o1}|1_{1}\rangle+c_{o2}|1_{2}\rangle\), with 1 and 2 denoting two single mode channel waveguides. Synchronous directional couplers can be represented by the following matrix  
\[X_{\theta}= \begin{pmatrix}\cos\theta&i\sin\theta\\ i\sin\theta&\cos\theta\end{pmatrix}  \tag{5}\]  
where \(\theta=\kappa L\), with \(\kappa\) the coupling coefficient and \(L\) the coupling (real or effective) length of the coupler. For single photon states, the standard algebra for a two-dimensional complex space can be applied, therefore the output state (or output 1-qubit) is given by the expression  
\[|L^{\prime}\rangle=(c_{o1}\cos\theta+ic_{o2}\sin\theta)|1_{1}\rangle+(ic_{o1} \sin\theta+c_{o2}\cos\theta)|1_{2}\rangle\equiv c_{1}|1_{1}\rangle+c_{2}|1_{2}\rangle \tag{6}\]For quantum information applications the usually required couplers are \(X_{\pi/4}\), that is, 3 dB (or 1:1) couplers, which correlate to the logic gate \(H_{c}=\sqrt{-iX}\), (circular Hadamard logic gate) and \(X_{\pi/2}\) corresponding to the usual Pauli's logic gate \(X\). On the other hand, phase shifters are also required. Their implementation can be obtained by using different strategies (change in refractive index, change in length or width of a single-mode channel guide and so on) and they can even be reconfigurable by using glass thermo-optic modulation.