In quantum mechanics, the conservation of information is indeed a fundamental principle. It is often referred to as the "reversibility" or "unitarity" of quantum dynamics. The conservation of information ensures that the evolution of a quantum system is reversible and that no information is lost or destroyed during the evolution.
This principle is derived from the basic axioms and mathematical formalism of quantum mechanics. The time evolution of a quantum system is governed by the Schrödinger equation, which is a unitary evolution equation. The unitary evolution preserves the norm of the quantum state, meaning that the total probability of all possible outcomes remains constant over time.
Mathematically, unitarity is expressed as the conservation of the inner product between quantum states. The inner product captures the overlap or similarity between two quantum states. The unitary evolution ensures that the inner product between any two quantum states remains conserved throughout the evolution.
The conservation of information is a consequence of unitary evolution because if information were lost or destroyed, it would imply a violation of the conservation of the inner product, leading to inconsistencies in the mathematical formalism of quantum mechanics.
It's worth noting that while the conservation of information is a foundational principle in quantum mechanics, it doesn't necessarily imply that all information can be easily accessed or extracted from a quantum system. Quantum systems can exhibit complex and entangled behavior, making it challenging to extract specific information in practice. However, the underlying principle of information conservation remains valid.