Entanglement Dynamics for Bell State Combinations under Various Noise Conditions

Quantum Entanglement is central to quantum information technology, yet it is highly sensitive to error and noise. In this work, we show quantitatively how noise modifies entangled states. Starting from the four Bell states, we quantify their entanglement under arbitrary superpositions and mixings, using partial tracing and von Neumann entropy in the first case and concurrence in the second. We also consider mixings of pure Bell states with noisy Bell states for depolarizing noise, phase damping, and amplitude damping. Finally, we consider the mixing of two noisy Bell states. We highlight counterintuitive situations where entanglement increases with increasing noise. Next, we introduce noise on only one of the qubits of the Bell state and obtain the Werner state in the case of depolarizing noise showing an alternative approach to the same interplay of noise with entanglement. We verify quantitatively that our results satisfy the Peres–Horodecki criterion. Our results contribute to a deeper understanding of entanglement dynamics in noisy environments and suggest potential applications in noise-assisted quantum protocols.