Part of that is questionable, because the state gets to set the means by which the quality of education is measured. However, a more objective measure is the extent of education; certainly, the introduction of national education has always produced a much greater literacy rate.
The thing is extent and execution are intrinsically linked. Meaning, unregulated markets mean the extent will be limited because people will just choose not to get educated. Which makes sense, if they themselves are uneducated they don't have perfect reasoning skills or future outlook.
A market relies on the participants having visibility and ability. They need to see the alternatives, understand them deeply, and have access to them. Turns out you can't do that in a bunch of markets, education being one of them. So, it can never be a true free market.
