# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 2 Time-Frequency Qubit States  
### Time-Frequency Cat State  
We denote \(|\Omega\rangle\) as the vacuum state. A single-photon state at frequency \(\omega\) in the spatial port \(a\) is denoted as \(\left|\omega\right\rangle_{a}=\hat{a}^{\dagger}(\omega)|\Omega\rangle\). The frequency cat state as introduced in [24] is defined as the superposition of a single photon into two different frequency Gaussian distributions (see Figure 1a):  
\[\left|\psi\right\rangle=N_{a\beta}(\alpha|\omega_{1}\rangle_{a}+\beta|\omega_{ 2}\rangle_{a})=N_{a\beta}(\alpha|0\rangle_{a}+\beta|1\rangle_{a}), \tag{1}\]  
where \(\left|\omega_{1}\right\rangle_{a}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\int d\omega \text{exp}(-(\omega-\omega_{1})^{2}/2\sigma^{2})|\omega\rangle_{a}\), \(\sigma\) is the frequency width of the spectral Gaussian distribution and the normalization of the state is given by \(1=N_{a\beta}^{2}(|\alpha|^{2}+|\beta|^{2}+2\text{Re}(\alpha\beta^{*})e^{-( \omega_{1}-\omega_{2})^{2}/2\sigma^{2}})\). This is a non-orthogonal qubit state, as their overlap is \({}_{a}\langle 0|1\rangle_{a}=\exp(-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2})\). Experimental proposals for manipulating such a state were proposed in [33; 34] with pulse shapers, and electro-optic modulators were used in the cascade. The results showed an operation fidelity close to 100%, but the successive optical elements drastically decreased the probability of single photon detection. The wave function of the time of the cat state is defined as  
\[|\psi_{t}\rangle=N_{\alpha\beta}(\alpha|t_{1}\rangle_{a}+\beta|t_{2}\rangle_{ a}). \tag{2}\]  
Note that for avoiding having a normalization constant depending on the coefficients \(\alpha,\beta\), and writing the wave function on an orthogonal basis, we can employ the Gram-Schmidt decomposition procedure. The normalized orthogonal basis \(|a\rangle\) and \(|b\rangle\) can be written as  
\[|a\rangle=|0\rangle,\ |b\rangle=N[|1\rangle-\langle 0|1\rangle|0\rangle], \tag{3}\]  
where \(N=1/\sqrt{1-r^{2}}\) and \(r=|\langle 0|1\rangle|\). The frequency Equation (1) (or time cat state Equation (2)) can be written in the orthogonal basis as  
\[|\psi\rangle=(\alpha+\beta\langle 0|1\rangle)|a\rangle+\frac{\beta}{N}|b\rangle, \tag{4}\]  
where we have now \(|(\alpha+\beta\langle 0|1\rangle)|^{2}+|\frac{\beta}{N}|^{2}=1\).  
The frequency entangled cat state, an EPR state, can be written as  
\[\left|\phi^{\pm}\right\rangle =N_{\text{EPR}}(\left|\omega_{1}\omega_{1}\right\rangle_{ab}\pm \left|\omega_{2}\omega_{2}\right\rangle_{ab}) \tag{5}\] \[\left|\psi^{\pm}\right\rangle =N_{\text{EPR}}(\left|\omega_{1}\omega_{2}\right\rangle_{ab}\pm \left|\omega_{2}\omega_{1}\right\rangle_{ab}) \tag{6}\]  
where \(N_{\text{EPR}}^{2}(2+2e^{-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2}})=1\). When \(\omega_{1}-\omega_{2}\gg\sigma\), we recover the normalization of an EPR state composed of orthogonal qubits \(N_{\text{EPR}}=1/\sqrt{2}\). The frequency cat state can be produced by an integrated optical wave guide [35] and bulk system [36]. The wave function of a temporal entangled EPR state Equation (5) has the same mathematical structure, and such a quantum state can be produced by quantum dots, for instance [37]. This type of quantum state has potential applications in quantum communications [38].