Entropy rates of low-significance bits sampled from chaotic physical systems

• Entropy of low-significance bits in digital measurements of chaos is examined.
• Low-significance bits yield a two-symbol partition with a corrugated structure.
• Corrugation at low-significance bits better approximates a generating partition.
• Entropy rate estimation using lower-significance bits requires longer block lengths.
• Considering only short block lengths can overestimate entropy of physical system.

We examine the entropy of low-significance bits in analog-to-digital measurements of chaotic dynamical systems. We find the partition of measurement space corresponding to low-significance bits has a corrugated structure. Using simulated measurements of a map and experimental data from a circuit, we identify two consequences of this corrugated partition. First, entropy rates for sequences of low-significance bits more closely approach the metric entropy of the chaotic system, because the corrugated partition better approximates a generating partition. Second, accurate estimation of the entropy rate using low-significance bits requires long block lengths as the corrugated partition introduces more long-term correlation, and using only short block lengths overestimates the entropy rate. This second phenomenon may explain recent reports of experimental systems producing binary sequences that pass statistical tests of randomness at rates that may be significantly beyond the metric entropy rate of the physical source.