f physicists Asher Peres and William Wootters had stuck to calling this quantum process ‘telepheresis’ when they first conceived of it in 19934, I doubt we’d be seeing headlines about it today. It was their co-author Charles Bennett who suggested instead ‘quantum teleportation’.You find some physicists in this field who act as if conservation of quantum information is the most important principle in all of physics. However, as Ball points out, the concept is not even defined.

Whatever it’s called, the process transfers the quantum state of one particle onto another, identical particle, and at the same time erases the state in the original. ...

So what exactly is being transmitted through entanglement alone?

This is a tricky question for quantum information theory in general: it is not obvious what ‘information’ means here. As with other colloquial words adopted by science, it is too easy to imagine we all know what we’re talking about. The 'stuff' transmitted by entanglement is neither information in the sense of Claude Shannon’s information theory (where it is quantified in terms of entropy, increasing as the 'message' gets more random), nor in the sense of an office memo (where information becomes meaningful only in the right context). Then what is it information about, exactly?

That issue, at the heart of quantum information theory, has not been resolved8, 9. Is it, for example, information about some underlying reality, or about the effects of our intervention in it? Information universal to all observers, or personal to each? And can it be meaningful to speak of quantum information as something that flows, like liquid in a pipe, from place to place? No one knows (despite what they might tell you). If we can answer these questions, we might be close finally to grasping what quantum mechanics means.

As far as I know, there are no experiments that have shown it to be ever conserved. There is not really any good theory for believing it to be conserved either, except for those who believe in time reversibility.

And "teleportation" is, as Ball says, just a misleading headline-grabbing term for some mundane quantum physics. Physics pretend that it is something magical, like Star Trek, but it is not.

Scott Aaronson has posted an argument for limits on information density:

Summarizing where we’ve gotten, we could say:This is plausible, but it is a very crude upper bound on classical information. Yes, he says info is physical in the sense that it must take up some space, or the energy needed to store it would be so large as to collapse into a black hole.any information that’s spatially localized at all, can only be localized so precisely. In our world, the more densely you try to pack 1’s and 0’s, the more energy you need, therefore the more you warp spacetime, until all you’ve gotten for your trouble is a black hole. Furthermore, if we rewrote the above conceptual argument in math—keeping track of all the G’s, c’s, h’s, and so on — we could derive a quantitativeboundon how much information there can be in a bounded region of space. And if we were careful enough, that bound would be precisely the holographic entropy bound, which says that the number of (qu)bits is at most A/(4 ln 2), where A is the area of a bounding surface in Planck units. ...

In summary, our laws of physics are structured in such a way thateven pure information often has “nowhere to hide”: if the bits are there at all in our description of the world, then they’re forced to pipe up and have a measurable effect. And this is not a tautology, but comes about only because of nontrivial facts about special and general relativity, quantum mechanics, quantum field theory, and thermodynamics. And this is what I think people should mean when they say “information is physical.”

But he is not saying that info is conserved, or giving equations of motion for info, or getting mystical about quantum info.