thermodynamics - Where does deleted information go?

I've heard that, in classical and quantum mechanics, the law of conservation of information holds.

I always wonder where my deleted files and folders have gone on my computer. It must be somewhere I think. Can anyone in principle recover it even if I have overwritten my hard drive?

Answer

Short Answer

The information is contained in the heat given off by erasing the information. Landauer' Principle states that erasing information in a computation, being a thermodynamically irreversible process, must give off heat proportional to the amount of information erased in order to satisfy the second law of thermodynamics. The emitted information is hopelessly scrambled though and recovering the original information is impossible in practice. Scrambling of information is what increasing entropy really means in plain English. Charles H. Bennett and Rolf Landauer developed the theory of thermodynamics of computation. The main results are presented in The thermodynamics of computation—a review.

Background

Erasure of information and the associated irreversibility are macroscopic/thermodynamic phenomena. At the microscopic level everything is reversible and all information is always preserved, at least according to the currently accepted physical theories, though this has been questioned by notable people such as Penrose and I think also by Prigogine. Reversibility of basic physical laws follows from Liouville's_theorem for classical mechanics and unitarity of the time evolution operator for quantum mechanics. Reversibility implies the conservation of information since time reversal can then reconstruct any seemingly lost information in a reversible system. The apparent conflict between macroscopic irreversibility and microscopic reversibilty is known as Loschmidt's paradox, though it is not actually a paradox.

In my understanding sensitivity to initial conditions, the butterfly effect, reconciles macroscopic irreversibility with microscopic reversibility. Suppose time reverses while you are scrambling an egg. The egg should then just unscramble like in a film running backwards. However, the slightest perturbation, say by hitting a single molecule with a photon, will start a chain reaction as that molecule will collide with different molecules than it otherwise would have. Those will in turn have different interactions then they otherwise would have and so on. The trajectory of the perturbed system will diverge exponentially from the original time reversed trajectory. At the macroscopic level the unscrambing will initially continue, but a region of rescrambling will start to grow from where the photon struck and swallow the whole system leaving a completely scrambled egg.

This shows that time reversed states of non-equilibrium systems are statistically very special, their trajectories are extremely unstable and impossible to prepare in practice. The slightest perturbation of a time reversed non-equilibrium system causes the second law of thermodynamics to kick back in.

The above thought experiment also illustrates the Boltzmann brain paradox in that it makes it seem that a partially scrambled egg is more likely to arise form the spontaneous unscrambling of a completely scrambled egg than by breaking an intact one, since if trajectories leading to an intact egg in the future are extremely unstable, then by reversibility, so must trajectories originating from one in the past. Therefore the vast majority of possible past histories leading to a partially scrambled state must do so via spontaneous unscrambling. This problem is not yet satisfactorily resolved, particularly its cosmological implications, as can be seen by searching Arxiv and Google Scholar.

Nothing in this depends on any non classical effects.