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Consider electron spin operator. It acts on Hilbert space $\mathbb{C}^2$. Next, electron position operator acts on space $\mathbb{L}^2$. Can we describe all electron features in one, "joint" Hilbert space? Certainly, both spaces can be combined into tensor product $\mathbb{C}^2 \otimes \mathbb{L}^2 $, but this looks like composition of the 2 subsystems. If this tensor product is legitimate construction, then the electron states can be divided into separable and entangled. Can particle be entangled with itself?

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It is certainly possible to entangle the position state of a particle with its spin state. This is exactly what a Stern-Gerlach apparatus does, producing quantum correlations between position and spin. If you do not observe the output of the Stern-Gerlach experiment directly (and so do not collapse the state vector), the output of the apparatus is precisely an entangled spin-position state.

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