# Can channels be generalized to linear maps from $\mathbb{C}^{n^k}$ to $\mathbb{C}^{n^k}$?

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.

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