The Replacement Principle

The entropy-replacement theorem is the result that makes Paper III a paper rather than a revision. In one line:

$\big[S(\rho_{R_A}) - S(\rho_{R_B})\big] - \big[H(P_A) - H(P_B)\big] = O_{L^2}\!\big(d^{-2}d_M^{-1}\big).$

The hard quantum disagreement, minus the easy classical disagreement, is small in mean square – and, crucially, one power of $d$ below the signal we are trying to measure, which scales as $\Theta(d^{-3}d_M^{-2})$. That margin is the whole point. It is not enough for the error to be small; it has to be small compared to the thing the paper claims to compute, or the claim drowns in its own correction. It is.

The shape of the proof is worth keeping on the page, because it is where the difficulty actually lives. You expand the true entropy around the bulk-marginal diagonal. The error $\Delta$ then splits into two pieces:

• $F_{\mathrm{off}}$ – the off-diagonal part, coming from the coherences the diagonal model throws away.
• $F_{\mathrm{diag}}$ – the diagonal-to-bulk part, coming from the difference between the actual reduced diagonal and the bulk-marginal diagonal we replace it with.

Both have to be shown to be $O(d^{-4}d_M^{-2})$ – one power of $d$ below the signal variance. $F_{\mathrm{off}}$ yields to a resolvent representation, a linear bound, and a fourth-moment estimate (the machinery of Appendix C.1–C.5). $F_{\mathrm{diag}}$ was the holdout. For a long time we could bound it only by splitting a covariance matrix into diagonal and off-diagonal blocks and arguing about the blocks – and the off-diagonal block is exactly where a numerical constant tried to sneak into the proof. That fight is the next two posts: the route that failed, and the route that worked.

What I want to flag here is conceptual, not technical. Before this theorem, the programme had a beautiful structure resting on an assumption. After it, the structure rests on a bound with a proof. The difference is the difference between “we believe” and “we have shown,” and earning that difference – for the Haar class, completely; for the product class, conditionally – is the entire content of Paper III. Everything else is bookkeeping around this inequality.