Is there a way to get the work item state date?

You can refer to the REST API for more details(https://jazz.net/wiki/bin/view/Main/ReportsRESTAPI)  on what properties can be used in your query in addition to the post you referred to to get the state name. For example, if I add [dayModified] to retrieve workitem 106's history for state as below, it will give me the day modified info:

https://localhost:9443/ccm/rpt/repository/workitem?fields=workitem/workItem[id='106']/(itemHistory/(state/name|dayModified))

If you're facile with Java, look at the JavaDoc of the Plain Java client libraries for IAuditableHandle.

public interface IAuditableHandle
extends IManagedItemHandle
	

Represents an auditable item. Auditable items exist in multiple states in a repository. There is a global notion of the current state of an auditable item; states other than the current are used for an audit trail.

Item handles and items can only be created or retrieved from a repository while the client is logged in. When the client logs out, all objects that they obtained while logged in must be considered invalid. It is unspecified whether these objects come back to life, so to speak, if the client logs back in.

I've used this to good effect to tease out the change date of a specific attribute so that the date can be used to answer before/after sort of question.

This post has a small bit of Java that got me started.

You could create a timestamp attribute to get the date when the workitem got to that state using calculated values scripts. And then query that field.