The minimum amount of time isn't something which is set in stone.
This will mainly depend on the owner's schedules and also on the type of household and whether the family is all engaged in spending daily quality time with the single pet bird or not.
I would say the very bare minimum of daily close interaction (this goes for both in or out of cage) throughout the day to be 2 to 3 hours.
RIP sweet Tito (Summer 2008 - January 17th 2013).
You are missed and never will be forgotten.