In matters relating to quantum or classical information theory, it is convenient to work with the simplest possible unit of information, the two-state system.
The meaning of the two-state system is very different in classical physics versus quantum physics. It is simple versus complex.
In classical information, this is a bit, commonly represented using one or zero (or true or false).
That is simple.
The quantum analog of a bit is a quantum bit, or qubit. Qubits encode a type of information, called quantum information, which differs sharply from "classical" information. For example, quantum information can be neither copied (the no-cloning theorem) nor destroyed (the no-deleting theorem).
The fact that a qubit cannot be copied or 'destroyed' are physical issues.
When you want to know the state of a quantum system consisting of qubits you have to perform a 'read' operation. In a logical sense such a 'read' operation changes the quantum state into a classical state. In a physical sense the same 'read' operation freezes the state of each qubit in either a zero or a one.
An important aspect of quantum information theory is entanglement, which imposes statistical correlations between otherwise distinct physical systems.
That is not wrong. The sentence should be:
An important aspect of quantum information theory is entanglement, which imposes statistical correlations between physical systems, having a common origin (source).
These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell test experiments.
This is a little like the chicken egg problem: What comes first the chicken or the egg.
At the start you almost know nothing except that you have an experiment, a reaction, which creates two particles (flying away in opposite directions).
The first time when you measure the spin of both particles the result is that one spin is up and the other one down. You repeat the experiment and the result is the same. You do that 1000 times: always the same.
The only difference between all the 1000 experiments is that when you consider one side: the outcome is random
What this means is that apperently the particles are correlated. This has nothing to do how and when the measurements are performed nor with any Bell inequality violation experiment. It is clearly a physical issue.
Understanding quantum teleportation requires a good grounding in finite-dimensional linear algebra, Hilbert spaces and projection matrixes.
Teleportation is a physical process. Understanding teleportation requires a understanding of the details of this physical process and does not require any form of mathematics.
A qubit is described using a two-dimensional complex number-valued vector space (a Hilbert space), which are the primary basis for the formal manipulations given below.
The whole issue is to what extend such a mathematical notation corresponds with the physical reality. Mathethematical notation in the sense all logical operations involved.
2 Protocol
3. Experimental results and records
4 Formal presentation
5 Alternative notations
6 Entanglement swapping
If Alice has a particle which is entangled with a particle owned by Bob, and Bob teleports it to Carol, then afterwards, Alice's particle is entangled with Carol's.
This sentence only make sense if 'entangled' is understood as correlated.
The problem is what exactly means: teleports.
A more symmetric way to describe the situation is the following: Alice has one particle, Bob two, and Carol one. Alice's particle and Bob's first particle are entangled, and so are Bob's second and Carol's particle: