The laws relating to chametz on Pesach include the prohibition of eating chametz, the obligation to get rid of chametz, and the prohibition of owning chametz.
However, it is not clear if these laws all go into effect at the same time. The prohibition of eating chametz and the obligation to get rid of it both begin a number of hours before the holiday starts. However, the Ra’avad is of the opinion that the prohibition of owning chametz applies only during the actual holiday, based on the verse, “No leaven shall be found in your houses for seven days” (Shemot 12:19). Rashi, in contrast, maintains that this prohibition too begins in the afternoon, at the same time as the other prohibitions.
There is also a difference of opinion as to the minimum amount (shiur) of chametz a person would have to possess in order to transgress the prohibition of ownership. The shiur in this case would seem to be an olive (kezayit). However, there is a general principle that even less than a shiur (chatzi shiur) is biblically prohibited (although the transgressor does not receive lashes). Some maintain that chatzi shiur is forbidden only when someone is doing something with the food (such as eating it), which makes it clear that this amount is significant to him (achshevei). However, if no action is involved (shev ve-al ta’aseh), as is the case with the prohibition of owning chametz, this principle might not apply. If so, owning a small amount of chametz (less than a kezayit) would be permitted on the biblical level.
Why should less than a shiur be prohibited? Shouldn’t the criterion, almost by definition, be the full shiur? One of the reasons for this stringency is the fear that someone will start by eating only part of a shiur, but will keep nibbling until, within a relatively short amount of time, he ends up eating an entire shiur. All that he ate combines together (mitztaref), and he is considered to have violated the prohibition from when he began eating. However, when we are dealing with a prohibition of ownership, even if someone ultimately acquires a full shiur, he will transgress only from the point of full acquisition onward, but not retroactively.