As a judge once said of pornography, " I know it when I see it."
In general the term "deep state" was recently coined by of all people Wikileaks founder Julian Assange but similar terminology predating this generally describes the "entrenched bureaucracy" which endures from administration to administration.
I like the term Deep State because it implies nefarious intentions. Therefore, by default, the connotations it invokes allows people's imaginations to run wild - and they should.
The ideological war being fought in the United States is one of class warfare in my opinion, but also can be viewed as a big government vs small government battle. Every day we are bombarded with flare ups depicting individual battles but at the end of the day each is only so much white noise that only serves to muddle the big picture.
We see half of the country protesting in the streets and carrying signs promoting big government. I want fewer choices they say. I want someone else to make life decisions for me they cry. Their signs don't use those exact words, but that is what they really mean.
Then we half the other half of the country saying get big government out of my life. I don't like those lies you feed us and the way you make everything you touch ten times worse. Leave me the hell alone.
So in one sense it doesn't really matter if a Deep State exists or not. If people believe dark forces are at work behind the scenes then it serves the greater good in my opinion. Distrusting big government is a very healthy state of mind.
If I was so inclined I could write several encyclopedias worth of information making the case for distrusting big government. It is a very easy case to make with mountains of evidence for backup. But it's really no more complicated than turning on your TV for five minutes or opening your front door and taking a good look around. The evidence is readily apparent for anyone to see.
Sometimes it takes a little nudge for people to open their eyes, however.