Exact Quantum Fisher Matrix Results for Distributed Phases Using Multiphoton Polarization Greenberger Horne Zeilinger States

In recent times, distributed sensing has been extensively studied using squeezed states. While this is an excellent development, it is desirable to investigate the use of other quantum probes, such as entangled states of light. In this study, we focus on distributed sensing, i.e., estimating multiple unknown phases at different spatial nodes using multiphoton polarization-entangled Greenberger Horne Zeilinger (GHZ) states distributed across different nodes.We utilize tools of quantum metrology and calculate the quantum Fisher information matrix (QFIM). However, the QFIM turns out to be singular, hindering the determination of quantum Cramer-Rao bounds for the parameters of interest. Recent experiments have contended with a weaker form of the Cramér-Rao bound, which does not require the inversion of the QFIM. It is desirable to understand how relevant these weaker bounds are and how closely they approach the exact Cramer-Rao bounds. We thus analyze the reason for this singularity and, by removing a redundant phase, obtain a nonsingular QFIM, allowing us to derive exact quantum Cramer-Rao bounds. Using the nonsingular QFIM, we show that the arithmetic average of the distributed phases is Heisenberg-limited. We demonstrate that the quantum metrological bounds can be saturated by projective measurements, enabling us to determine the Fisher information matrix (FIM), which is also singular. We then show how this singularity can be resolved.