# High-Accuracy Entanglement Detection via a Convolutional Neural Network with Noise Resistance
## 2 Methods
### Noise Models  
Pauli noise is a comprehensive noise model that includes Pauli matrices \((\sigma_{x},\sigma_{y},\sigma_{z})\), representing quantum bit flipping, simultaneous bit and phase flipping, and phase flipping, respectively. In quantum systems, this type of noise can cause errors in the state of quantum bits, ranging from simple flipping to more complex errors. In a two-qubit system, qubit Pauli noise is defined as [33]  
\[\epsilon(\rho)=(1-\epsilon)\rho+\epsilon\sum_{k=1}^{16}p_{k}(\sigma_{1,i} \otimes\sigma_{2,j})\rho(\sigma_{1,i}\otimes\sigma_{2,j})^{\dagger}. \tag{4}\]  
In this model, \(\rho\) represents the initial density matrices of the two-qubit quantum state. \(\epsilon\) is a parameter characterizing the intensity of the noise, with a value ranging between 0 and 1. \(\sigma_{1,i}\otimes\sigma_{2,j}\) denotes the combination of Pauli operations (including an additional \(2\times 2\) identity matrix \(\mathbf{I}\)) applied to the two qubits, where \(\sigma_{1,i}\) and \(\sigma_{2,j}\) are Pauli operators acting on the first and second qubit, respectively. \(k\) represents the probability distribution for the \(k^{th}\) combination of operations, satisfying \(\sum_{k=1}^{16}p_{k}=1\) to ensure the completeness of probabilities. To simplify our analysis, we opted for the basic formula,  
\[\epsilon(\rho)=(1-\epsilon)\rho+\epsilon(\mathbf{I}\otimes\sigma_{2,x})\rho( \mathbf{I}\otimes\sigma_{2,x})^{\dagger}, \tag{5}\]  
and it is worth noting that other forms can adhere to the same logical framework as well.  
Depolarizing noise causes a quantum bit to gradually lose its specific state, transitioning toward a completely mixed state. Depolarizing noise makes the information in a quantum bit more random. The formula for two-qubit depolarizing noise is  
\[\epsilon(\rho)=(1-\epsilon)\rho+\frac{\epsilon\mathbf{I}}{4}, \tag{6}\]  
and the definitions of each parameter are consistent with those previously mentioned.  
Dephasing noise primarily affects the phase of a quantum bit rather than its amplitude. Phase damping does not change the probabilities of the qubit being in \(\ket{0}\) and \(\ket{1}\) but causes changes in their phase relationships. In a two-qubit system, dephasing noise is given by  
\[\epsilon(\rho)=(1-\epsilon)\rho+\frac{\epsilon}{2}((\sigma_{2}\otimes \mathbf{I})\rho(\sigma_{2}\otimes\mathbf{I})^{\dagger}+(\mathbf{I}\otimes \sigma_{z})\rho(\mathbf{I}\otimes\sigma_{z}))^{\dagger}. \tag{7}\]  
Amplitude-damping noise is often associated with energy loss, such as the transition of a quantum bit from an excited state to a ground state. This type of noise is particularly common in quantum optical systems, where quantum bits lose energy through interactions with the environment. Amplitude-damping noise is defined with the Kraus operator. \(\mathbf{E}_{0}=\ket{0}\bra{0}+\sqrt{1-\epsilon}\ket{1}\bra{1}\) and \(\mathbf{E}_{1}=\sqrt{\epsilon}\ket{0}\bra{1}\). Its operator-sum representation can be expressed as  
\[\epsilon(\rho)=\mathbf{E}_{0}\rho\mathbf{E}_{0}^{\dagger}+\mathbf{E}_{1}\rho \mathbf{E}_{1}^{\dagger}. \tag{8}\]  
The four noise models enable us to simulate the detrimental effects of noise by mixing the original quantum state with its counterparts that have undergone diverse Paulitransformations, where the noise intensity is governed by the parameter \(\varepsilon\). By meticulously adjusting both the \(\varepsilon\) value and the probabilities \(p_{k}\) associated with each specific combination of Pauli operations \(\sigma_{1,i}\otimes\sigma_{2,j}\), we can thoroughly analyze the evolution of the quantum state under a given noise scenario. These noise models mentioned are then applied directly to the density matrices, enabling us to quantify the influence of noise, and, consequently, simulate its implications on the performance of entanglement detection utilizing CNNs. This approach provides insights into the robustness of entanglement detection in the presence of various noise scenarios.