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Recently, I've been interested in the restriction of operators on complex Hilbert spaces to the real numbers. This seems like an interesting way to quantify how "complex" a vector is with respect to some reference basis. My idea is that with this quantification we may approximate Hilbert spaces with Euclidean spaces in scenarios where complex numbers do not effect the dynamics of a calculation (which will cut the number of parameters in half in some cases).

Consider a tensor product $\mathcal H \otimes \mathcal K$ of two Hilbert spaces $\mathcal H$ and $\mathcal K$ of finite dimensions $m$ and $n$ respectively. Let $\{h_1,...,h_m\}$ and $\{k_1,..,k_n\}$ be orthonormal reference bases of $\mathcal H$ and $\mathcal K$ respectively and $\{h_1 \otimes k_1,h_1 \otimes k_2,...,h_m \otimes k_n\}$ the reference orthonormal basis of $\mathcal H \otimes \mathcal K$. I am looking at the existence of unitary operators $U: \mathcal H \otimes \mathcal K \to \mathcal H \otimes \mathcal K$ which are orthogonal with respect to the reference basis and preserve the seperability of vectors.

In other words, given two separable vectors $\alpha \otimes \beta$ and $\alpha' \otimes \beta'$ where $\alpha,\alpha' \in \mathcal H$ and $\beta,\beta' \in \mathcal K$, does there exist an operator $U$ whose matrix is orthogonal in the reference basis such that $$U (\alpha \otimes \beta)=\alpha' \otimes \beta'$$ and if not, what are the conditions on say, $\alpha$ and $\alpha'$ which would make the existence possible. The existance is clear when each of the vectors are real linear combinations of the basis vectors but it becomes much less clear in the case of $\mathbb C$ linear combinations.

I ask this in the context of orthogonal transformations on the Hilber space $\mathcal H$. For given vectors $\alpha,\alpha' \in \mathcal H$, there may not be an orthogonal map $P: \mathcal H \to \mathcal H$ such that $P\alpha=\alpha'$, however, the introduction of an ancilla space $\mathcal K$ may allow for some sort of "catalyst" like relation where there are vectors $\beta,\beta' \in \mathcal K$ and a unitary $U$ on the joint space such that the transformation is possible.

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