Alice Gross, 14, was found in the River Brent in west London in September 2014.
Days later the body of Latvian builder Arnis Zalkalns was found nearby.
The inquest into her death will look into whether failures by the government and the police contributed to her death and will examine why Zalkalns was allowed to live unchecked in the UK.
Reading from a prepared statement at the inquest at the Royal Courts of Justice, Alice's mother Rosalind Hodgkiss said the family wanted to "establish whether or not the systems for monitoring foreign offenders and cross-border sharing of information are robust".
She added: "We appreciate that they may have changed significantly, but we remain stunned that a foreign national with a conviction for murder was not monitored, or even known about in any way.
"This has destroyed much of our faith in our country's ability to protect its citizens.
"The Home Office and the police forces nationwide should be doing everything they can to ensure that this should not be allowed to happen again."
Ms Hodgkiss described her daughter as "lively and popular" as well as "witty, sharp and academic".
She said Alice was a keen songwriter and it was "impossible to convey the devastation her death had caused to the family".
She added: "We have many unanswered questions. We will never know exactly what happened on that day."
Alice went missing on 28 August 2014 after leaving for a walk along the canal. Her body was found a month later following what the Met Police called the largest inquiry since the 7/7 bombings in 2005.
Pathologist Dr Ashley Fegan-Earl told the inquest her body had been discovered in the River Brent, tied into a foetal position and weighted down by bricks, logs and a bicycle wheel.
He concluded she had been sexually assaulted and asphyxiated, probably by being prevented from breathing by a larger person pressing on her torso.
Zalkalns was the chief suspect in Alice's death and the Metropolitan Police and Crown Prosecution Service (CPS) both said he would have been charged with her murder had he not died.
Zalkalns was convicted of murdering his wife in Latvia in 1998.
The mother of a schoolgirl who police believe was killed by a foreign convicted murderer said she was stunned he was not being monitored.