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First let's set some terminology. Recall that a quantum channel is in particular a linear map $\Phi : \text{L} ( \mathcal{X}) \rightarrow \text{L} ( \mathcal{Y})$ where $\mathcal{X}$ and $\mathcal{Y}$ are complex finite dimensional vector spaces. For simplicity, let $\mathcal{X} = \mathcal{Y}$ and let $\dim \mathcal{X} = n$. Then any linear map in $\text{L}(\mathcal{X})$ can be represented as $n \times n$ matrix. Therefore, $\Phi$ can be represented as a $n^2 \times n^2$ matrix. Therefore, one can think of $\Phi$ as a map from $\mathbf{C}^{n^2}$ to $\mathbf{C}^{n^2}$.

The following question intrigued me while working on a project:

In the quantum information literature, have people considered generalization of linear maps $\Phi$ that can be thought of maps from $\mathbf{C}^{n^k}$ to $\mathbf{C}^{n^k}$ for $k > 2$. Are there any applications of such maps in quantum information, quantum machine learning etc?

Knowing the answer to this question would help me figure out if I need to consider the case $k > 2$ for my research.

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  • $\begingroup$ So, something like a CP map acting on tensors? $\endgroup$
    – Rammus
    Nov 22, 2022 at 10:39
  • $\begingroup$ This is user22511 from a different account: Yes. $\endgroup$
    – user82261
    Nov 22, 2022 at 11:42

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