CHARACTERIZING ENTANGLEMENT IN COMBINATIONS OF BELL STATES THROUGH SUPERPOSITION AND MIXING: AN INCREASE IN ENTANGLEMENT ON INTRODUCING DEPOLARIZING, PHASE DAMPING, AND AMPLITUDE DAMPING NOISE

This paper investigates how quantum entanglement and non-locality behave when two-qubit Bell states are combined under three operational regimes: pure superposition, noiseless mixing, and asymmetric mixing of one noisy and one pure state. The authors systematically analyze all sixteen possible combinations of the four standard Bell states. To quantify entanglement, they use von Neumann entropy for pure superpositions and Wootters' concurrence for mixed states. Furthermore, they evaluate operational quantum non-locality using the Horodecki criterion to determine when states can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality.

Noise is modeled using three physically motivated channels: Depolarizing (DP), Phase Damping (PD), and Amplitude Damping (AD). For DP and PD, the authors apply a global, collective decoherence model across the two-qubit Hilbert space, reflecting scenarios where qubits interact with a shared environment in Noisy Intermediate-Scale Quantum (NISQ) architectures. Conversely, AD is modeled as independent local decay on each qubit to capture the asymmetric, energy-dissipating nature of spontaneous emission. Because Bell states possess an X-state structure that is preserved by these channels, the authors derive simplified analytical expressions for both concurrence and the Horodecki parameter without requiring full matrix diagonalization.

For pure superpositions, the entanglement depends critically on the relative phase between the superposed states and the mixing weight. In the mixed and noisy regimes, the analysis reveals a central and counterintuitive result: introducing noise can actually increase concurrence in specific parameter regimes across all three channels. Additionally, the boundary between mathematical entanglement and CHSH-violating non-locality varies drastically by channel. Under Phase Damping, any surviving entanglement guarantees a CHSH violation for identical and same-bases mixtures. Under Depolarizing noise, a large Bell-local region requires a high mixing probability to recover non-locality at maximum noise. For Amplitude Damping, certain mixtures exhibit a non-monotonic Horodecki parameter where concurrence decreases monotonically, yet CHSH violation capacity is lost at intermediate noise and restored at high noise due to maximal damping repurposing the noisy branch.