The Born Rule in quantum mechanics relates the probability of obtaining a particular measurement outcome to the squared modulus of the wave function describing the system. It does not imply that information is lost in a measurement process.
In quantum mechanics, the act of measurement generally causes the system to undergo a collapse of its wave function to one of the eigenstates corresponding to the measured observable. This collapse is often described as "projecting" the system onto the measured state. The probabilities of different measurement outcomes are given by the squared moduli of the coefficients in the superposition of eigenstates present in the wave function prior to measurement.
While the measurement process causes a specific outcome to be observed, it does not mean that information is lost. The post-measurement state is known, and the probabilities for different outcomes are determined by the Born Rule. However, the exact details of the pre-measurement state are generally not recoverable from the post-measurement state alone.
Regarding CPT symmetry, it is a fundamental symmetry of quantum field theories that combines charge conjugation (C), parity inversion (P), and time reversal (T). This symmetry implies that the amplitudes for a process and its time-reversed counterpart are the same. However, it does not imply that the amplitudes for alternative processes are zero. It simply states that the probabilities for the original process and its time-reversed counterpart are equal.
In summary, the Born Rule does not imply information loss in measurement. The measurement process causes the collapse of the wave function to a specific outcome with probabilities determined by the squared moduli of the wave function coefficients. CPT symmetry relates the amplitudes for a process and its time reversal, but it does not imply that the amplitudes for alternative processes are zero.