Dynamic evolution of quantum Fisher and skew information under decoherence in three-qubit X-states

Quantum metrology leverages quantum effects such as squeezing, entanglement, and other quantum correlations to boost precision in parameter estimation by saturating the quantum Cramér-Rao bound, which can be achieved by optimizing quantum Fisher information (QFI) or Wigner-Yanase skew information (SI). QFI evaluates the sensitivity of a quantum state to infinitesimal variations in a parameter, determining the maximum precision of its estimate. In contrast, SI measures the non-commutativity between a quantum state and an observable, illustrating the perturbative effect of that measurement. This work provides analytical expressions for quantum Fisher and skew information in a general three-qubit X-state and examines their evolution under phase damping, depolarization, and phase-flip decoherence channels. To illustrate the validity of our method, we investigate their dynamics for a three-qubit Greenberger-Horne-Zeilinger (GHZ) state subjected to various memoryless decoherence channels. Closed-form expressions for QFI and SI are derived for each channel. By comparing these metrics with the entanglement measure of concurrence, we demonstrate the impact of decoherence on measurement precision for quantum metrology. Our results indicate that phase damping and phase-flip channels generally allow for better parameter estimation compared to depolarization. This study provides insights into the optimal selection of noise channels for enhancing precision in quantum metrological tasks involving multi-qubit entangled states.