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Is a direct sum of two Hilbert spaces a subspace of their tensor product space? If not is there any relation between those two?

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  • $\begingroup$ $\mathbb C \oplus \mathbb C=\mathbb C^2$ but $\mathbb C \otimes \mathbb C=\mathbb C$. $\endgroup$ – Jochen Aug 30 '17 at 7:21
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There is no canonical embedding of the direct sum of Hilbert space into their tensor product. However, we can of course come up with an embedding of the direct sum into the tensor product such as $$ \phi_{y_1,y_2}:H_1 \oplus H_2 \to H_1 \otimes H_2\\ (x_1,x_2) \mapsto x_1 \otimes y_2 + y_1 \otimes x_2 $$ where $y_1,y_2$ are non-zero elements of $H_1,H_2$ respectively.

Perhaps a more natural relationship between direct sums and tensor products is that $$ H \otimes \Bbb C^n \cong \overbrace{H \oplus H \oplus \cdots \oplus H}^n = \bigoplus_{k=1}^n H $$ There is also a "distributive property", which is to say that $$ (H_1 \oplus H_2) \otimes H_3 \cong (H_1 \otimes H_3) \oplus (H_2 \otimes H_3) $$

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