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Two fermions cannot share the same quantum state.

But two electrons can be entangled. Entangled electrons share the same quantum state.

Thus a contradiction.

Where is my error?

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    $\begingroup$ Your error is in the sentence "Entangled electrons share the same quantum state". $\endgroup$ – WillO Oct 29 '15 at 1:38
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The claim "Entangled electrons share the same quantum state" is not correct. In an entangled state there is no well-defined notion of the states of the individual components, this is the very definition of an entangled state:

A composite state $\chi\in\mathcal{H}_1\otimes\mathcal{H}_2$ is called entangled, if it cannot be written as $\chi=\psi\otimes\phi$ for $\psi\in\mathcal{H}_1,\phi\in\mathcal{H}_2$.

The composite states forbidden by the Pauli exclusion principle, however, are states of the form $\psi\otimes\psi\in\mathcal{H}$ for the same $\psi\in\mathcal{H}$, where both particles have the same space of states $\mathcal{H}$.

Thus, there is no contradiction.

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I am no expert, but I think looking up Quantum entanglement on Wikipedia will tell you that the entangled particles are in correlated states, but that is not the same as sharing the same state. 

Also, as a side issue, two electrons cannot have the same 4 quantum numbers, if they are bound to an atom, but this does not apply to free electrons.

 

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