Evolution of Entanglement Witness of Dicke State under Noise and Error Mitigation

The experimental verification of multipartite entangled states is essential for advancing quantum information processing. Entanglement witnesses (EWs) provide a widely used and experimentally accessible approach for detecting genuinely multipartite entangled states. In this work, we theoretically derive the entanglement witness for the four-qubit Dicke state and experimentally evaluate it on two distinct IBM 127-qubit Quantum Processing Units (QPUs), namely ibm\_sherbrook and ibm\_brisbane. A negative expectation value of the witness operator serves as a sufficient condition for confirming genuine multipartite entanglement. We report the maximum (negative) values of the witness achieved on these QPUs as $-0.178 \pm 0.009$ and $-0.169 \pm 0.002$, corresponding to two different state preparation protocols. Additionally, we theoretically investigate the effect of various noise channels on the genuine entanglement of a four-qubit Dicke state using the Qiskit Aer simulator. We show the behavior of the EW constructed under the assumption of Markovian and non-Markovian amplitude damping and depolarizing noises, bit-phase flip noise, and readout errors. We also investigate the effect of varying thermal relaxation time on the EW, depicting a bound on the $T_1$ time required for successful generation of a Dicke State on a superconducting QPU.