Conservation Laws for Meteorologically Useful Quantum Information

It has been shown that the eigenvalues of the Quantum Fisher Information Matrix (QFIM) serve as generalized entanglement witnesses, and have long been known to yield a lower bound for the single parameter Quantum Cramer Rao Bound. A natural question to then ask is what operations may change these eigenvalues, and when are they conserved quantities. Here, we answer these questions by deriving a set of fundamental conservation laws for the QFIM, and establish which operations break this invariance. Our results directly recover the notion that entanglement via "squeezing" is an irreducible resource under passive linear optical transformations, and our conditions for non-conservation may be used to describe all entanglement generating protocols. As an example, we demonstrate the application of our work to classic squeezing protocols such as one axis twisting and two axis counter twisting. We highlight that these results are only dependent on the underlying geometry of the Hilbert space and unitary evolution. We apply conservation laws to multiparameter sensing and the optimization of measurement-dependent classical fisher information, and to generalizations of the Quantum Regret.