No Time Asymmetry from Quantum Mechanics

With CPT-invariant initial conditions that commute with CPT-invariant final conditions, the respective probabilities (when defined) of a set of histories and its CPT reverse are equal, giving a CPT-symmetric universe. This leads me to question whether the asymmetry of the Gell-Mann–Hartle decoherence functional for ordinary quantum mechanics should be interpreted as an asymmetry of {\it time} .