fix purity phrasing

Author: lpawela

docs/src/asymptotic.md

@@ -111,7 +111,7 @@ using Symbolics
 page = 2d / (d^2 + 1)
 asymptotic(page, d, 5)
 # Output: 2/d - 2/d^3 + 2/d^5
-# Leading 2/d → purity approaches 1/d (maximally mixed)
+# Leading 2/d: purity is O(1/d); maximally mixed would be exactly 1/d
 ```
 
 ```julia

docs/src/qi_helpers.md

@@ -69,13 +69,15 @@ using IntU, Symbolics
 page_purity = 2n / (n^2 + 1)
 asymptotic(page_purity, n, 5)
 # Output: 2/n - 2/n^3 + 2/n^5
-# Leading term: 2/n → subsystem approaches maximally mixed state
+# Leading term: 2/n (same O(1/n) scaling as maximally mixed purity 1/n)
 ```
 
 > [!NOTE]
-> For large subsystems, the leading term $2/n$ shows the purity approaches
-> $1/n$ (the maximally mixed value), confirming that Haar-random pure states
-> are nearly maximally entangled — the core of Page's theorem.
+> For large subsystems, the purity is order $1/n$:
+> $\mathbb{E}[\mathrm{tr}(\rho_A^2)] = 2/n + O(1/n^3)$ for equal bipartitions.
+> A maximally mixed $n$-dimensional state has purity exactly $1/n$, so this
+> result indicates highly mixed (and nearly maximally entangled) subsystems,
+> not equality with the maximally mixed value.
 
 ## See Also
 

txt/manuscript.tex

@@ -864,9 +864,11 @@ \subsection{Asymptotic expansions}
 # Output: 2/n - 2/n^3 + 2/n^5
 \end{lstlisting}
 
-The leading term $2/n$ shows that the subsystem approaches the maximally mixed
-state (purity $1/n$) as the dimension grows, which is the core of Page's
-theorem: a random subsystem is nearly maximally entangled. The subleading
+The leading term $2/n$ shows that the subsystem purity is of order $1/n$ as the
+dimension grows. For comparison, a maximally mixed $n$-dimensional state has
+purity exactly $1/n$, so Page's law indicates highly mixed (and thus nearly
+maximally entangled) reduced states rather than equality with the maximally
+mixed value. The subleading
 corrections $-2/n^3 + \cdots$ quantify the finite-size deviations from
 maximal entanglement. Such expansions automate the derivation of scaling laws
 where manual simplification of the underlying Weingarten sums would be