Generalized Fubini-Study Metric and Fisher Information Metric

TL;DR

This paper introduces a measurable, gauge-invariant Fubini-Study metric for mixed quantum states that captures quantum uncertainty, satisfies the quantum Cramer-Rao bound, and generalizes to a family of $oldsymbol{oldsymbol{ extalpha}}$ metrics.

Contribution

It develops a new measurable Fubini-Study metric for mixed states, relates it to Fisher information, and introduces a generalized family of $oldsymbol{ extalpha}$ metrics satisfying key quantum bounds.

Findings

The new FS metric satisfies the quantum Cramer-Rao bound.

On the Fisher information space, the dynamical phase is zero.

The $ extalpha$ metric generalizes the FS metric and reduces to Fisher information for $ extalpha=1$.

Abstract

We provide an experimentally measurable local gauge $U(1)$ invariant Fubini-Study (FS) metric for mixed states. Like the FS metric for pure states, it also captures only the quantum part of the uncertainty in the evolution Hamiltonian. We show that this satisfies the quantum Cramer-Rao bound and thus arrive at a more general and measurable bound. Upon imposing the monotonicity condition, it reduces to the square-root derivative quantum Fisher Information. We show that on the Fisher information metric space dynamical phase is zero. A relation between square root derivative and logarithmic derivative is formulated such that both give the same Fisher information. We generalize the Fubini-Study metric for mixed states further and arrive at a set of Fubini-Study metric---called $\alpha$ metric. This newly defined $\alpha$ metric also satisfies the Cramer-Rao bound. Again by imposing the monotonicity condition on this metric, we derive the monotone $\alpha$ metric. It reduces to the Fisher information metric for $\alpha=1$.