Infanticide is legally defined as the unlawful killing of a child aged under 12 months, by its biological mother, when at the time the "balance of the mother's mind was disturbed".
The details of the Court of Appeal's judgement have only been made public on Friday after the conclusion of the case.
In the findings, the appeal court said the original convictions were quashed after it examined three grounds of appeal: the conduct of the prosecution; evidence by an expert witness; and handling by the judge.
"That is because the serious errors in the conduct of this trial which began with the prosecution's misconduct were compounded by the expert witness and the judge."
The woman's defence team argued that the judge, Donna McColgan, had failed to ensure a fair trial.
They pointed to flaws in evidence given by an expert witness, Dr Richard Bunn, and they argued that Richard Weir KC's (Crown counsel) handling of Dr Bunn before and during the trial amounted to misconduct.
During the 2023 trial two psychiatrists gave evidence.
Dr Murphy, who was an expert in infanticide, said the woman was suffering from moderate to severe depression which substantially impaired her rational decision-making and self-control and believed the defence of infanticide was open to the defendant.
Dr Murphy also told the court that the mother in this case appeared to be the victim of coercive control by her partner, the father of her children.

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