B Quantum definition of information

Information is not a well-defined physical concept. It is an informal, more intuitive but subjective, way of talking about the objective physical (and statistical) concept of entropy.

 

There is a technical definition of "information", but I don't think that you actually need that technical definition to understand this issue.

In both classical mechanics and quantum mechanics, physics is reversible at the microscopic level. For our purposes, that just means that distinct initial states lead to distinct final states. If at time [itex]t_1[/itex] you know that a system is in state [itex]A[/itex], then the equations of physics will allow you (in principle) to compute the state it will be in at a later time, [itex]t_2[/itex]. Call that state [itex]B[/itex]. Reversibility just means that you run the equations backwards: If you know that the system was in state [itex]B[/itex] at time [itex]t_2[/itex], then you can (again, in principle) use the equations of physics to compute the state at the earlier time, [itex]t_1[/itex]. So this is the intuitive sense in which information is never lost: in principle, you can always recover any information about the past.

In practice, it seems like things are not reversible. If you put an ice cube on a hot metal plate, it will predictably melt into a puddle of water. So your equations allow you to compute the future state from the past state. On the other hand, if you start with a plate holding a puddle of water, you can't figure out that it used to be an ice cube. Hot water cooling off leads to the same final state as ice melting. So melting seems irreversible. But in classical physics, at least, the microscopic details--the positions and momenta of the water molecules--will be different depending on whether it was once an ice cube. The information about the past was not truly lost, it just became inaccessible--it's stored in the details of the motions of microscopic particles.

Now, for black holes radiating through Hawking radiation, it seems irreversible. You can throw anything into a black hole--a car, a potted plant, etc., the final state of the black hole is unchanged. When the black hole radiates away, there seems to be no record left, even at the microscopic level, of what went into the black hole. Since all the physics involved in black hole radiation---General Relativity, quantum mechanics---are reversible at the microscopic level, it's a puzzle how you can have irreversible processes arise from reversible processes. So the question is: Is black hole formation and evaporation really irreversible? If so, how did irreversibility arise from reversible laws of physics? If not, then where did the information go when the black hole evaporated?