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With reference to this PDF discussing Superdense coding, it is mentioned that there are two qubits A and B whose superposition gives the system $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$. It is said that Alice has A while Bob has B. What are the individual qubits A and B?

Since the system is entangled, can there exist two qubits A and B such that their superposition is $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$ ? If the answer is no, how is $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$ actually shared between Alice and Bob (or maybe how is it realized in practise)? Is it just that A and B cannot be written using the standard basis?

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Is it just that A and B cannot be written using the standard basis?

Correct. Entangled qubits are not separable. You can't factor them into a form like $(a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)$.

You can get a per-qubit density matrix by tracing out the other qubit, but the resulting density matrix doesn't capture the correlation between the two qubits. It just looks like a maximally mixed state.

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