Can quantative easing be summarized as "The state prints cash to pay its debt to private sector"?
Altough you are correct saying that the state "own" central banks, they should be independent from their government. This gives freedom to monetary policymakers from direct political or governmental influence in the conduct of policy.

Given that, I would say that your interpretation is a bit exaggerated and maybe biased somehow.

Quantitative easing (QE) is a monetary policy in which a central bank purchases government securities or other securities from the market in order to increase the money supply. This is made with newly created bank reserves, but doesn't mean that new currency (bills and coins) has been printed. In theory, with more liquidity, the domestic demand would increase and eventually promote economic growth. Then, the excess of liquidity is absorbed by the growth and inflation would remain low.

The problem is if the economic growth don't happen, there will be inflation.

Using a more informal language, it's correct to say that the government is creating money in the present, hoping that this fresh money will increase economic growth in the future.