# Entangled electrons seem impossible

Two fermions cannot share the same quantum state.

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

Where is my error?

• Your error is in the sentence "Entangled electrons share the same quantum state". – WillO Oct 29 '15 at 1:38

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 $\mathcal{H}$.