According to the no-cloning theorem it is impossible to create an independent and identical copy of an arbitrary unknown quantum state. We cannot delete any quantum information as well. All changes in time of the state vector in quantum mechanics are described by the action of unitary operators. Accordingly, there must be an operator performing a deleting operation. The operator must be a zero matrix in order to nullify quantum information totally in all cases. But a zero matrix is not a uninaty or hermitian matrix. Therefore there is no such unitary operator that can delete information.
This might be proven in another way. Let us imagine the double-slit thought experiment where interference exists when we do not know about the system and interference does not when we know about the system. Assume we have a storage where the data is stored and the experiment is being conducted with knowing about the system. Suppose we destroy the storage. What does the screen in the experiment show us? Quantum mechanics tells us that there must not be interference. Should it appear after the data is destroyed? Since the wave function has collapsed it cannot be restored. If there is a chance to delete the information in the experiment, it means that the wave function must go back to the initial state and show us interference, which is a contradiction.
Based on the foregoing, we will consider the quantum eraser experiment. In that experiment information is neither erased nor disappeared. It is being changed. That is the key point. We increase entropy. If there is a 50 per cent chance to get interference then the entropy = 1 (max value). The same with a spin. If we change spins of elementary particles, for example in the Stern–Gerlach experiment with different axes measurements, we do not delete the information about the states of particles, we increase the entropy. Changing does not equal deleting.
Conclusion is that we cannot delete information; we can only change it by increasing its entropy.
References:
1. W.K. Wootters and W.H. Zurek, "A Single Quantum Cannot be Cloned", Nature 299 (1982)
2. Griffiths, D. J. Introduction to Quantum Mechanics (2nd ed.). Essex England: Pearson Education (2004).
3. Eibenberger, Sandra; et al. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10000 amu
4. Gerlach, W.; Stern, O. (1922). "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld"
5. Walborn, S. P.; et al. (2002). "Double-Slit Quantum Eraser". Phys. Rev. A. 65 (3)
