Fisher-Rao metric

In plain terms

Two probability distributions can be close together or far apart, and statisticians have a standard way to say how far, which asks how easily a sample could tell them apart. If moving from one distribution to the other barely changes what you'd observe they're close, and if it changes the observations a lot they're far. Fisher information is the local version of that idea, the rate at which observations respond to a small change, and the Fisher-Rao metric turns it into a geometry, a way of measuring lengths and angles on the space of distributions.

Quantum states have a geometry of their own. The space of states of a finite quantum system carries a distance, the Fubini-Study metric, which is the only one that treats every state the same way and respects the symmetries of the theory. A measurement turns a state into a probability distribution over its outcomes, so you can ask whether the two geometries agree.

They do, on the directions the measurement can see. Moving a state so that only the sizes of its components change, and not their phases, moves its outcome distribution, and the quantum distance moved equals the statistical distance moved. Moving the phases is invisible to the measurement, so those directions have quantum length and no statistical length. The gap between the two is the information a fixed measurement leaves on the table, which is why the best measurement of a parameter reaches the quantum limit exactly when the state moves in a modulus-only direction.

In CSD

The corpus builds the state space of a finite quantum system as a Kahler manifold, with the Fubini-Study form, metric and volume proved on it rather than assumed, and the Born weights appear on that manifold as the moment map of the torus that rotates the phases of the coordinates. This result reads the moment map as a map between geometries and finds that it carries the Fubini-Study inner product of two modulus-only directions to the Fisher-Rao inner product of their images, with constant one in the corpus's normalisation, and that it sends no such direction to zero, so it's an isometric embedding of the modulus-only directions into the simplex's tangent space.

The value for the programme is that the metric side of its geometry is checked against a structure statisticians already own, and the value outside it is direct. The result is written so that the quantum information library Physlib can take it as it stands, and it was built for that library's Fisher information work, with the vector-level statement placed beside Physlib's own Fisher-Rao file and the manifold statement following from it in an affine chart.

Mathematically

For a unit vector psi with no vanishing coordinate the Born weights p_i = |psi_i|^2 form a point of the open simplex, and a direction u moves them by dp_i = 2 Re(conj(psi_i) u_i). Along a torus-horizontal direction, one for which every conj(psi_i) u_i is real, the Fisher-Rao inner product of the displacements of u and any v equals four times the real inner product of u and v (fisherRaoInner_bornDeriv); that is the Fubini-Study inner product once u and v are also tangent to the unit sphere, and the projective statement builds the tangency in by projecting the direction off psi first (horizontalLift, fisherRaoInner_bornDeriv_normalize). Along that projected direction the coordinate readout's Fisher information is at most fsInnerHom psi u u, which for unit psi is the quantum Fisher information 4(|u|^2 - |<psi, u>|^2) (fsInnerHom_self_of_norm_eq_one), with equality exactly when the projected direction is torus-horizontal (fisherInfo_bornDeriv_horizontalLift_le, _eq_iff), which is the Braunstein-Caves bound for that measurement.

On the manifold the same statement reads fsMetric x u v = fisherRaoInner (toOpenSimplex x) (momentDeriv x u) (momentDeriv x v) on the regular stratum for u in the horizontal space, the Fubini-Study orthogonal complement of the torus orbit (fsMetric_eq_fisherRaoInner), and the differential is injective on that space at every point of the regular stratum (momentDeriv_injOn_horizontalSpace). The constant is one because the corpus normalises the Fubini-Study metric so that the qubit's state space is the unit sphere, where the metric is itself the quantum Fisher information (four times the Bengtsson-Zyczkowski metric), and the identification of the corpus's metric with that formula on lifts is its own theorem (fsMetric_eq_fsInnerHom). The map called the moment map here is the Born-weight map; the Hamiltonian of the torus action for the corpus's form is twice its pairing with the angles, so the moment map in the Hamiltonian sense is twice this map, and the constant one is for the weights. Every constant declared in the closure of the result depends on the foundational triple of axioms only.

The name

Calyampudi Radhakrishna Rao (1920-2023) was born in Hadagali in what's now Karnataka, trained at Andhra University and Calcutta, and joined the Indian Statistical Institute under Mahalanobis in the early 1940s. His 1945 paper in the Bulletin of the Calcutta Mathematical Society, written when he was twenty-five, contains both the Cramer-Rao bound and the observation that Fisher information defines a Riemannian metric on a family of distributions, which is the metric that carries his name with Fisher's. He led the Institute's research and training for decades, moved to the University of Pittsburgh and then Pennsylvania State University, and received the National Medal of Science in 2002.

Ronald Fisher's part of the name is his information measure of 1922, and his own entry tells his story.

Related entries

• Fisher informationThere's a floor under every error bar, classical or quantum, and Fisher information is what sets it.
• Moment mapThe Born probabilities are coordinates, and the moment map reads them straight off the geometry of the space of quantum states, with no measurement postulate consulted.
• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• Born ruleQuantum probabilities aren't a law of nature in CSD, as they're the sizes of regions in the space of states, measured with a ruler nobody was free to choose.
• Kahler formOn the space of quantum states, distance, area and a quarter-turn coming from the complex numbers lock together, and that lock is what makes the volume of the space nobody's choice.