Geometric Invariants of Quantum Metrology

TL;DR

This paper uncovers a conservation law for the Quantum Fisher Information Matrix, showing its spectrum remains invariant under certain symmetries, thus revealing fundamental limits on quantum metrological sensitivity and state classification.

Contribution

It introduces a novel invariance property of the QFIM under Lie algebra-generated dynamics, linking geometric invariants to quantum metrological resources.

Findings

QFIM spectrum invariance under Lie algebra dynamics

Quantum incompatibility fixed by symmetry group

Invariant statistical distances and volumes under evolution

Abstract

We establish a previously unexplored conservation law for the Quantum Fisher Information Matrix (QFIM) expressed as follows; when the QFIM is constructed from a set of observables closed under commutation, i.e., a Lie algebra, the spectrum of the QFIM is invariant under unitary dynamics generated by these same operators. Each Lie algebra therefore endows any quantum state with a fixed "budget" of metrological sensitivity -- an intrinsic resource that we show, like optical squeezing in interferometry, cannot be amplified by symmetry-preserving operations. The Uhlmann curvature tensor (UCT) naturally inherits the same symmetry group, and so quantum incompatibility is similarly fixed. As a result, a metrological analog to Liouville's theorem appears; statistical distances, volumes, and curvatures are invariant under the evolution generated by the Lie algebra. We discuss this as it relates to the quantum analogs of classical optimality criteria. This enables one to efficiently classify useful classes of quantum states at the level of Lie algebras through geometric invariants.