For random product (low-entanglement) bulk states, the disagreement falls off only as the negative one-half power – far slower than the Haar case. The exponent and prefactor are derived analytically. Stated conditional on the product-class replacement principle, because product states have tiny probability entries that make the entropy perturbation delicate.
For random product (low-entanglement) bulk states, the disagreement falls off only as the negative one-half power – far slower than the Haar case.
Why conditional, and on what. This law is stated conditional on the product-class replacement principle. The Haar replacement (Result I) is proved unconditionally because Haar bulk states spread probability evenly; the entropy perturbation analysis behaves. Product states do the opposite: almost all probability lives on a few entries, almost none on the rest. The entropy function’s derivative blows up as a probability goes to zero, so the clean bound that works for spread-out distributions does not obviously survive when they are concentrated.
We could have papered over this. The product-class numerics match the prediction at the precision the rest of the paper holds itself to. But matching numerics is the standard we explicitly refused on Result II, and we will not apply it selectively. So Theorem 3 wears the word conditional – in the abstract, on this card, and in the discussion – until somebody (us or anyone else) supplies the small-mass control that turns it unconditional.
The integer exponent gap (Result IV) between this law and Result II’s Haar law is the same in either case. Its claim survives the conditional intact.