Tensor product in an exponent

Question

I stumbled over the expression $\mathcal{H}=(\mathbb{C}^d)^{\otimes k}$. But I don't understand what this this means in context to the dimension of the Hilbert space $\mathcal{H}$.

Answer 1

If $V$ is a vector space, $V^{\otimes k}$ denotes the tensor product of $k$ copies of $V$. For example $$V^{\otimes 2}=V\otimes V$$ and $$V^{\otimes 3}=V\otimes V\otimes V$$

Since the dimension of a tensor product is the product of the dimensions of the components $$\text{dim }V^{\otimes k}=(\text{dim }V)^k$$

In your case, the dimension of $\mathcal{H}$ will be $d^k$.