What information is retained on the boundary of a black hole? If an object were to be pulled into a black hole, I’ve been told its information would be retained on the boundary.
But will that be just the information of the actual state it was in at the time? Or the information of its actual state and also any other possible state it could have been in?
 A: First consider what is meant by the claim that information is stored on the surface defined by the event horizon of a black hole.
Anything that falls into a black hole, whether it be a macroscopic object, or a single quantum particle, the information describing these things is said to be smeared on a two-dimensional sphere, described by the event horizon. The mechanism whereby this information is stored over this surface is complex, and I'm not sure that all physicists agree on how this occurs. It is said that the information is encoded on the surface as a hologram.

But will that be just the information of the actual state it was in at the time? Or the information of its actual state and also any other possible state it could have been in?

It seems that you are talking about how the information describing a quantum particle is stored on the event horizon, where earlier you spoke about "objects" in general.
Whether the particle is in a pure state (superposition) or a if it's in a definite state at the time, you'd think the mechanism for the encoding and storage of information is the same. The problem becomes when we consider that this information is not stored in such a way so as that the information can be meaningfully decoded. This suggests that the original information could be lost.
Also, if we consider Hawking radiation, if a particle described by a pure quantum state (as opposed to a mixed quantum state), i.e., one that could be written as a superposition of other eigenstates,
were to fall into a black hole, then the
radiation that is emitted will be associated with a mixed state.
This mixed state will provide no direct information about the original pure state, and because this information is not preserved, this implies that unitarity is violated.
It is one of the axioms of quantum mechanics that a state evolves unitarily in accordance with the Schrodinger equation, which
means that quantum information stored in a system is preserved over time.
