No. The SLA Breakdown "breakdown field name" seems to have its values restricted via the following....
OOB the ONLY options available are Assigned To and Assignment Group and I believe that's by intent. The vital essence of the feature is to determine the WHO of SLA time consumption. And since SLAs are so often started, stopped, and paused based on states, I'm not certain what a state breakdown would provide.
You can STILL get state breakdown information of a sort though...
Here's a quick and dirty one I did that sums the durations for each state based on the Metric Instance table.
