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In this paper, by Adam Nahum et al., the authors trivially states that "For Clifford dynamics all Rényi entropies are equal ... " which is not trivial to me.

Is there a paper or lecture notes on why this is true or a good explanation?

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Let me report here a brief demonstration that I used to convince myself. The R'enyi entropies are \begin{equation*} S_\mathrm{A}^\alpha = \frac{1}{1-\alpha} \log_2 \text{Tr} (\rho_\mathrm{A}^{\alpha}) \end{equation*} In a nutshell, this is a consequence of the reduced density matrix $\rho_{\mathrm{A}}$ being a stabilizer itself, and hence it can be written a sum over the elements of the stabilizers group $\mathcal{S}_{\mathrm{A}}$. Here $\mathcal{S}_{\mathrm{A}}$ indicates the stabilizer group of all stabilizers that act trivially (identity) on the B part as defined in Appendix C of this paper. We begin with \begin{equation} \rho_\mathrm{A} = \frac{|\mathcal{S}_\mathrm{A}|}{2^{|\mathrm{A}|}} \frac{1}{|\mathcal{S}_{\mathrm{A}}|} \sum_{s\in \mathcal{S}_\mathrm{A}}s. \end{equation} By squaring the density matrix, we obtain: \begin{equation} \rho_{\mathrm{A}}^2 = \frac{1}{2^{2 |\mathrm{A}|}} \sum_{s \in \mathcal{S}_{\mathrm{A}}} \sum_{s' \in \mathcal{S}_{\mathrm{A}}} s s' = \frac{1}{2^{2 |\mathrm{A}|}} |\mathcal{S}_{\mathrm{A}}|\sum_{s \in \mathcal{S}_{\mathrm{A}}} s = \frac{|\mathcal{S}_{\mathrm{A}}|}{2^{|\mathrm{A}|}} \rho_{\mathrm{A}} \end{equation} In the second equality we utilize the fact $\mathcal{S}_{\mathrm{A}}$ is a group, hence the product between its elements generate elements within the same group, and this happens exactly $|\mathcal{S}_{\mathrm{A}}|$ times. It is evident that in general $\rho_{\mathrm{A}}^{\alpha} = (|\mathcal{S}_{\mathrm{A}}| / 2^{|\mathrm{A}|})^{\alpha - 1} \rho_{\mathrm{A}}$. By substituting this into the expression for the R'enyi entropy, we obtain \begin{align*} S_{\mathrm{A}}^{\alpha} & = \frac{1}{1-\alpha} \log_2 \text{Tr} \bigg[ \bigg(\frac{|\mathcal{S}|}{ 2^{|\mathrm{A}|}} \bigg)^{\alpha - 1} \rho_{\mathrm{\alpha}} \bigg] \\ & = \frac{1}{1-\alpha} \log_2 \bigg(\frac{|\mathcal{S}_\mathrm{A}}{ 2^{|\mathrm{A}|}} \bigg)^{\alpha - 1} \text{Tr} \rho_{\mathrm{A}} \\ & = |\mathrm{A}| - \log_2|\mathcal{S}_{\mathrm{A}}| \\ \end{align*} which is exactly the result we achieved in this paper and similar as this paper.

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In stabilizer states (which are the states produced by clifford dynamics) the entanglement spectrum is flat, i.e., all the non-zero Schmidt coefficients are equal. Starting from there, it is straightforwards to check that all Renyi entropies are equal, namely $\log D$, where $D$ is the number of non-zero Schmidt coefficients.

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