The claim, which is a precursor to a lawsuit, also alleges the woman was raped repeatedly before giving birth last December.
A former licensed practical nurse at Hacienda Healthcare has been charged with sexually assaulting the 29-year-old woman, who has been in long-term care since age 3 after suffering a near-drowning.
She gave birth to a boy at the facility on Dec. 29. Employees said they had no idea she was pregnant.
The claim seeks a $25 million settlement for the victim and $10 million for each of her parents within 60 days or the lawyers will take the case to court.
The office of Gov. Doug Ducey said Wednesday night that it had received the 55-page notice of claim and would be reviewing it while the Arizona Department of Health Services said it doesn't comment on pending or ongoing litigation.
The Attorney General's Office said the state is being represented by outside council in the case and they had no further comment.
According to medical records cited in the claim, the woman was "violently and repeatedly raped" while living at Hacienda and a doctor who examined her on the day she gave birth noted that she could have been pregnant before.
"At a minimum, there were repeated violations of (the victim) from the scarring," John Micheaels, an attorney representing the victim and her family, told Phoenix TV station KPNX.
The surprise birth triggered reviews by state agencies, highlighted safety concerns for patients who are severely disabled or incapacitated and prompted the resignations of Hacienda's chief executive and one of the victim's doctors.
Investigators say Nathan Sutherland's DNA matched a sample from the woman's newborn boy, who is being cared for by her family.
Sutherland, 37, has pleaded not guilty to charges of sexual abuse and abuse of a vulnerable adult.
Hacienda fired Sutherland after his arrest. He has since given up his nursing license.
According to the notice of claim, the victim's mother had specifically requested "female-only staff," saying her daughter was at risk for exploitation.
The claim says Sutherland provided unsupervised care for the victim over 1,000 times, including more than 800 times overnight.
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