SSAS - Inferred Dimension Attributes - are they necessary?
I'm not sure I understand your question fully, so please clarify if necessary.

If you want to query the cube to get persons by state, you should add a Referenced relationship between Person and State, using Case as the Intermediate dimension.


Select the empty box at the intersection of Person and State and click on the button
Select Referenced from the relationship type dropdown
Select Case as the Intermediate dimension
Selecting the matching attributes in the two attribute dropdowns
Click OK.


Deploy and process the cube, and it should work the way you want.  Then you can get rid of the (State - Person) dimension.