Sadly not possible using searchstates. There have been different workarounds, one of which is to resort to SQL. For example:
FROM news_c_mungo ncm
LEFT JOIN news_a na
ON ncm.cs_attrid = na.id
where na.name = 'CopiedDescription'
and stringvalue is NULL
returns any News_C assets having a stringvalue attribute called "CopiedDescription" that is null.