The no-hiding theorem[1] states that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the correlation between the system and the environment.
This is a fundamental consequence of the linearity and unitarity of quantum mechanics.
The no-hiding theorem is robust to imperfection in the physical process that seemingly destroys the original information.
In 2011, the no-hiding theorem was experimentally tested[2] using nuclear magnetic resonance devices where a single qubit undergoes complete randomization; i.e., a pure state transforms to a random mixed state.
Subsequently, the lost information has been recovered from the ancilla qubits using suitable local unitary transformation only in the environment Hilbert space in accordance with the no-hiding theorem.
This experiment for the first time demonstrated the conservation of quantum information.
be an arbitrary quantum state in some Hilbert space and let there be a physical process that transforms
denotes the fact that one may augment the unused dimension of the environment Hilbert space by zero vectors.
The proof of the no-hiding theorem is based on the linearity and the unitarity of quantum mechanics.
The original information which is missing from the final state simply remains in the subspace of the environmental Hilbert space.
Also, note that the original information is not in the correlation between the system and the environment.
The no-hiding theorem provides new insights to the nature of quantum information.
However, quantum information cannot be completely hidden in correlations between a pair of subsystems.
In physics, conservation laws play important roles.
It can neither increase nor decrease without coming in contact with an external system.
If we consider the whole universe as a closed system, the total amount of energy always remains the same.
In the classical world, information can be copied and deleted perfectly.
But the no-hiding theorem is a more general proof of conservation of quantum information which originates from the proof of conservation of wave function in quantum theory.
It may be noted that the conservation of entropy holds for a quantum system undergoing unitary time evolution and that if entropy represents information in quantum theory, then it is believed then that information should somehow be conserved.
Since the wave function contains all the relevant information about a physical system, the conservation of wave function is tantamount to conservation of quantum information.