In theory, yes. But when you include hedging costs and taxes, the link becomes less direct. The cleanest way to get long volatility is by purchasing ATM straddles.
With a put, you primarily pay for directionality with hedged upside risk: you don't lose your house if the stock moons. While it's true volatility is a component, that's a side effect of the hedging since your counter party takes on volatility risk.
https://www.optionseducation.org/strategies/all-strategies/s...
This is not quite all the way there, but close enough. Basically, you do something analogous to 1 = 1/2 + 1/4 + 1/8 + ...