a persistence barcode
When you watch topological features appear and disappear as a data set is scanned across all scales, you want a compact picture of the whole story. A persistence barcode is exactly that picture: a stack of horizontal line segments, one per topological feature, drawn against the scale axis. Each bar starts where its feature is born and ends where it dies, so a glance tells you which features are long-lived (real structure) and which are blips (noise). The persistence diagram is the same information plotted as points instead of bars.
Concretely, after running persistent homology on a filtration, each homology class in degree k yields an interval [b, d) where b is its birth scale and d its death scale (possibly +infinity for a feature that never dies). The barcode in degree k is the multiset of all these intervals drawn as horizontal bars; the equivalent persistence diagram plots each as a point (b, d) in the plane above the diagonal y = x, where distance from the diagonal equals the lifespan d - b. The structure theorem for persistence modules guarantees this decomposition into intervals is unique, so the barcode is a complete invariant of the persistence module — it loses nothing. To compare two barcodes you use the bottleneck distance (the smallest epsilon such that the diagrams' points can be matched within epsilon, allowing unmatched points to pair with the diagonal) or the Wasserstein distance.
Barcodes matter because they turn a continuous, scale-dependent topological signal into a finite, comparable summary that you can feed to statistics or machine learning — vectorizations like persistence landscapes and persistence images make them usable as features. The honesty points: a barcode records lifespans, not locations, so two very different shapes can share a barcode; a single point off the diagonal in the diagram can correspond to a class with no canonical geometric representative; and bars near the diagonal are exactly the noise that the stability theorem promises will move only a little under small data perturbations — which is why you usually read the long bars and discard the short ones, but 'short' is a threshold you must justify, not a law.
For the noisy circle, the H_1 barcode shows one long bar plus a few short ones; in the persistence diagram that long bar is a single point far above the diagonal, while the noise sits in a cloud hugging the diagonal. The bottleneck distance to the barcode of a clean circle is small, certifying that the noisy sample is topologically 'a circle' at the scales that matter.
Long bars sit far from the diagonal (real features); noise clusters near it.
A barcode is a complete invariant of the persistence module but not of the underlying space: it records when features live and die, never where they are, so distinct geometries can produce identical barcodes. Treating short bars as noise is a modeling choice with a threshold you must justify, not a theorem.