## 7 Joint Spectrum
In Figures 15 and 16, the JSA describes strong spectral correlation between the signal and idler photons because of energy and momentum conservation constraints. In other words, if the signal (idler) photon is detected to herald the production of the idler (signal) one, the heralded photon is not in a pure state but in a mixed state. This is not suitable for the Linear Optics Quantum Computing (LOQC) protocol which is based on frequency-uncorrelated photons, allowing direct heralding of single photons in pure state. Generally, in photonic quantum computing, the error rates of logic gates  
Figure 16: (**a**) Pump envelope function, \(|\alpha(v_{s},v_{i})|^{2}\), governed by energy conservation. (**b**) Phase-matching function, \(|\Phi(v_{s},v_{i})|^{2}\), governed by momentum conservation. (**c**) Joint spectral amplitude, \(|F(v_{s},v_{i})|^{2}\), as expressed in Equation (24). \(\lambda_{p}=1064\) nm, \(\Delta\lambda_{p}=3\) nm, and a 5 cm-long MNF with the 950 nm-diameter are used for numerical simulation. Figure 15: (**a**) Pump envelope function, \(|\alpha(v_{s},v_{i})|^{2}\), governed by energy conservation. (**b**) Phase-matching function, \(|\Phi(v_{s},v_{i})|^{2}\), governed by momentum conservation. (**c**) Joint spectral amplitude, \(|F(v_{s},v_{i})|^{2}\), as expressed in Equation (24). \(\lambda_{p}=725\) nm, \(\Delta\lambda_{p}=1\) nm, and a 12 cm-long MNF with the 615 nm-diameter are used for numerical simulation. increase with photon state impurity which decreases the Hong-Ou-Mandel (HOM) dip visibility [55]. In the heralded process, the detection of one photon in a twin-photon source indicates that the other photon is projected into a single-photon state. Photon-pairs created in SFWM are usually frequency-entangled due to energy and momentum conservation constraints [56,57]. Therefore, in this case, the heralding process projects the twin photon into a mixed state [58]. Even though this entanglement can be eliminated by using spectral and spatial filters, it degrades the production rate of created photons and heralding efficiency [59]. Here, we show that frequency-entangled states can be factorable states by controlling pump bandwidth and group delay. In the JSA, \(|F(v_{s},v_{i})|\) given in Equation (24), \(\alpha(v_{s},v_{i})\) always exhibits an anti-diagonal slope in frequency space, since the maximum value is governed by the relation \(v_{s}+v_{i}=0\). On the other hand, the slope of \(\Phi(v_{s},v_{i})\) is determined by the condition \(\tau_{s}v_{s}+\tau_{i}v_{i}=0\), so that the angle of the slope in frequency space depends on the signs of the group velocities of pump, signal, and idler. We now present mathematical condition for generating factorable states by using Gaussian approximation, \(\text{Sinc}\left[\frac{\tau_{s}v_{s}+\tau_{i}v_{i}}{2}\right]\simeq\text{Exp }\left[-r\frac{(\tau_{s}v_{s}+\tau_{i}v_{i})^{2}}{4}\right]\) with \(r=0.193\). Consequently, JSA can be given as [60,61]  
\[ \begin{split} F(v_{s},v_{i})&=\text{Ce}^{-i\frac{ \tau_{s}v_{s}+\tau_{i}v_{i}}{2}}\text{Exp}\left[-\frac{(v_{s}+v_{i})^{2}}{2 \sigma_{p}^{2}}\right]\text{Sinc}\left[\frac{\tau_{s}v_{s}+\tau_{i}v_{i}}{2} \right],\\ &\simeq\text{Ce}^{-i\frac{\tau_{s}v_{s}+\tau_{i}v_{i}}{2}}\text {e}^{-\left[\frac{\tau_{s}^{2}}{2}\left(\frac{1}{\sigma_{p}^{2}}+\frac{\tau_{ s}^{2}}{2}\right)\right]}\text{e}^{\left[-\frac{\tau_{s}^{2}}{2}\left(\frac{1}{ \sigma_{p}^{2}}+\frac{\tau_{s}^{2}}{2}\right)\right]}\text{e}^{\left[-v_{s}v_{ i}\left(\frac{1}{\sigma_{p}^{2}}+\frac{\tau_{s}\tau_{i}}{2}\right)\right]}.\end{split}  \tag{25}\]  
In order to make a factorable state in the JSA, the frequency entangled term between the signal and idler, \(\text{Exp}\left[-v_{s}v_{i}\left(\frac{1}{\sigma_{p}^{2}}+\frac{\tau_{s}\tau_ {i}}{2}\right)\right]\), should be eliminated.