Fisher information

In plain terms

Suppose you're estimating something, a frequency, a field strength or a phase, from noisy data. Better analysis shrinks your error bar up to a point, so you can ask whether there's a floor, a best possible precision that no cleverness in processing can beat.

There is, and Fisher information is its name. It measures how sharply the distribution of your data responds when the underlying parameter shifts, so responsive data can pin the parameter down tightly and sluggish data can't. Turn the information upside down by taking its reciprocal and you have a hard lower bound on the spread of any unbiased estimate, meaning any estimate that's right on average.

The quantum version adds one more optimisation. Instead of considering only the ways of processing the data from a fixed measurement, it ranges over all possible measurements, so it yields the ultimate floor physics allows for a given probe state.

In CSD

Fisher information lives in the corpus's metrology strand, which the documentation classifies carefully as an adjacent arm and not a step in the reconstruction. It consumes the formalised machinery of states, the Born rule and dynamics to state precision bounds and the protocols that meet them, and it contributes nothing to deriving quantum mechanics and doesn't pretend to.

The reason to carry it at all is that a formalisation claiming practical reach should be able to state the standard quantum limit, the improved Heisenberg scaling that entanglement buys, and the gap between them, and the corpus can.

The strand does touch the geometry at one point. The corpus proves that the natural distance on the space of quantum states agrees with the statistician's distance on the outcome probabilities along every direction a fixed measurement can see, which is the Fisher-Rao metric entry's result, and that agreement is what a Physlib collaboration on Fisher information asked the corpus for.

Mathematically

In the classical setting, Fisher information is the variance of the score of a parametrised family, and the Cramer-Rao inequality bounds any unbiased estimator's variance below by its inverse.

In the quantum setting, maximising over measurements gives the quantum Fisher information and the Braunstein-Caves bound, the precision floor for a given parametrised family of states before any choice of apparatus.

For the coordinate readout of a pure state the corpus proves the bound itself and its equality case: along the projective direction of change (the component of the change orthogonal to the state, so that moving along the state itself counts for nothing) the classical Fisher information of the readout is at most the quantum Fisher information (fisherInfo_bornDeriv_horizontalLift_le), with equality exactly when that direction changes the moduli of the coordinates and not their phases (fisherInfo_bornDeriv_horizontalLift_eq_iff).

The name

Ronald Fisher (1890-1962) was born in London, read mathematics at Cambridge, and did his transformative work not at a university but at Rothamsted Experimental Station from 1919, where crop trials posed the problems that produced experimental design, analysis of variance and maximum likelihood. Chairs followed, the Galton Chair at University College London and then Cambridge.

The record has a dark seam that the name of his London chair advertises. Fisher was a lifelong, institutionally active eugenicist, and his statistics were in significant part built in service of that programme. Late in life he took tobacco- industry consulting money while publicly disputing the link between smoking and cancer. Both facts belong in any honest biography, next to the fact that modern statistics is unimaginable without him.

Related entries

• Ramsey interferometryTwo brief nudges separated by a long stretch of doing nothing measure a frequency more sharply than any amount of continuous pushing, and every atomic clock runs on this trick.
• phase estimationPhase estimation, the workhorse subroutine of quantum computing, carries a guarantee, as if you read the dial at the nearest tick you're right at least four divided by pi squared of the time, a little over forty percent, however fine the dial.
• 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.
• amplitude estimationPoint phase estimation at the rotation inside amplitude amplification and the success probability itself becomes measurable, quadratically faster than sampling could ever learn it.
• Robertson uncertainty relationRobertson's inequality is the precise form of the uncertainty principle, and in CSD the spreads it bounds are the spreads of quantities genuinely distributed over the underlying state space and not merely abstract expectation values.
• Fisher-Rao metricThe natural way to measure distance between quantum states agrees with the statistician's way to measure distance between the probability distributions they produce, exactly and with no constant left to fix.