Maybe start with linear algebra?
Here is a plain-ASCII Slashdot-friendly explanation using linear algebra, kept short:
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Quantum teleportation looks mysterious, but the linear algebra makes it clear why it is not just "sending info the normal way."
Let the unknown state be
|psi> = a|0> + b|1>
with a and b unknown. Alice and Bob share an entangled Bell pair, say
|B> = (|00> + |11>) / sqrt(2)
The combined 3-qubit system is
|psi> (tensor) |B>
If you expand this product in the Bell basis on Alice's side, you get:
|psi>|B> = 1/2 * [ |Phi+> (a|0> + b|1>)
+ |Phi-> (a|0> - b|1>)
+ |Psi+> (b|0> + a|1>)
+ |Psi-> (b|0> - a|1>) ]
Each Bell state on Alice's side corresponds to Bob's qubit being some *linear transform* of the original |psi>. Specifically, Bob's qubit is one of:
I |psi>, Z |psi>, X |psi>, XZ |psi>
(all 2x2 Pauli matrices: I, X, Z).
Alice measures in the Bell basis. Her 2-bit measurement result just tells Bob *which* of the four linear transforms he must invert. He applies the corresponding Pauli matrix, restoring exactly:
|psi> = a|0> + b|1>
Notice what this means:
* Alice never learns a or b.
* Only entanglement allows |psi> to be rewritten in the Bell basis like this.
* The classical bits do not contain the quantum amplitudes; they only index which linear operator Bob must undo.
* The original state is destroyed by Alice's measurement, so no cloning occurs.
Teleportation is simply a linear-algebra identity: entanglement lets you rewrite the tensor product in a basis where Bob's qubit equals |psi> up to one of four known matrices. The classical message just tells him which matrix to invert.