so it follows that

$$ \left| H \left( \sum_i p_i \rho_i \right) - \sum_i p_i H(\rho_i) \right| \le P \log(m) + |P \log(P)| + \max |\log(q_i)| \\ \le C(\log(m) + 2 \log(C)). $$

□

The idea is that for a random tensor network state, we will split up the background state as a superposition of states which are maximally entangled along the two minimal cuts and then use Lemma 4.13. This approach formalizes an argument sketched in [6].

To state our result, we introduce a new function: For vectors of positive numbers $p \in \mathbb{R}^{d_1}$, $q \in \mathbb{R}^{d_2}$, we let

$$ H^*(p, q) := \sum_{i,j} p_i q_j \min \left( \log \left( \frac{1}{p_i} \right), \log \left( \frac{1}{q_j} \right) \right). $$

A key tool we will need is the continuity of the entropy: If $\rho, \sigma \in \mathcal{P}_\le(\mathcal{H})$ are quantum states on a Hilbert space $\mathcal{H}$ of dimension $d$ with $T(\rho, \sigma) \le \frac{1}{e}$, then the Fannes–Audenaert inequality states that

$$ |H(\rho) - H(\sigma)| \le T(\rho, \sigma) \log d - T(\rho, \sigma) \log(T(\rho, \sigma)). \quad (4.44) $$

Let $\eta$ be the function defined by $\eta(x) = x \log x$. Then, if we write $\text{spec}(\rho) = \{p_j\}_{j=1}^d$ and $\text{spec}(\sigma) = \{q_j\}_{j=1}^d$ for Eq. (4.44) one can actually show

$$ |H(\rho) - H(\sigma)| \le \sum_{j=1}^{d} |\eta(p_j) - \eta(q_j)| \le T(\rho, \sigma) \log d - T(\rho, \sigma) \log(T(\rho, \sigma)). $$

We will use this to show continuity of $H^*$ as well. Consider $\rho_i, \sigma_i \in \mathcal{P}_\le(\mathcal{H}_i)$ with $\dim(\mathcal{H}_i) \le d$ for $i = 1, 2$. We let $\text{spec}(\rho_i) = \{p_{i,j}\}_{j=1}^{d_i}$ and $\text{spec}(\sigma_i) = \{q_{i,j}\}_{j=1}^{d_i}$. For real numbers $x_i, y_i$ we have

$$ |\min(x_1, y_1) - \min(x_2, y_2)| \le |x_1 - x_2| + |y_1 - y_2|. $$

Then, we see that

$$ \begin{aligned} & |H^*(\text{spec}(\rho_1), \text{spec}(\rho_2)) - H^*(\text{spec}(\sigma_1), \text{spec}(\sigma_2))| \\ & \le \sum_{j,k} \left| p_{1,j} p_{2,k} \min \left( \log \left( \frac{1}{p_{1,j}} \right), \log \left( \frac{1}{p_{2,k}} \right) \right) - q_{1,j} q_{2,k} \min \left( \log \left( \frac{1}{q_{1,j}} \right), \log \left( \frac{1}{q_{2,k}} \right) \right) \right| \\ & \le \sum_{j,k} \left| p_{1,j} p_{2,k} \log p_{1,j} - q_{1,j} q_{2,k} \log q_{1,j} \right| + \left| p_{1,j} p_{2,k} \log p_{2,k} - q_{1,j} q_{2,k} \log q_{2,k} \right|. \end{aligned} $$
We may estimate the first term by

$$
\begin{align*}
\sum_{j,k} |p_{1,j}p_{2,k} \log p_{1,j} - q_{1,j}q_{2,k} \log q_{1,j}| & \\
&\le \sum_{j,k} p_{2,k} |p_{1,j} \log p_{1,j} - q_{1,j} \log q_{1,j}| - \sum_{j,k} |p_{2,k} - q_{2,k}| q_{1,j} \log q_{1,j} & \\
&\le \sum_j |\eta(p_{1,j}) - \eta(p_{2,j})| + \|\rho_2 - \sigma_2\|_1 H(\sigma_1) & \\
&\le T(\rho_1, \sigma_1) \log d - T(\rho_1, \sigma_1) \log(T(\rho_1, \sigma_1)) + 2T(\rho_2, \sigma_2) \log d &
\end{align*}
$$

The second term may be estimated in similar fashion. We conclude that if
$T(\rho_1, \sigma_1) \le \varepsilon \le e^{-1}$ and $T(\rho_2, \sigma_2) \le \varepsilon \le e^{-1}$

$$
|H^*(\mathrm{spec}(\rho_1), \mathrm{spec}(\rho_2)) - H^*(\mathrm{spec}(\sigma_1), \mathrm{spec}(\sigma_2))| \le 6\varepsilon \log d - 2\varepsilon \log(\varepsilon). \quad (4.45)
$$

We will now show that we can approximate the entropy as would be
expected from Theorem 4.12, if we make some additional assumptions (which
in particular are satisfied if the state along each edge is a copy of n states,
$\phi_e = \phi_{e,0}^{\otimes n}$).

**Corollary 4.14.** Let $\rho$ be a random tensor network state satisfying the same assumptions as in Theorem 4.12, and assume additionally that the $\nu_{\phi_{\gamma_{A,i}}}$ have uniformly exponentially decaying tail probabilities and $\varepsilon(n) = \mathcal{O}(n^{-\gamma})$ for $\gamma > 10$. Assume that for each edge $e = (xy)$, the bond dimension is $D_e = 2^{\mathcal{O}(n)}$ and $\mathrm{tr}[\phi_{e,x}^2] \le 2^{-\Omega(n)}$. Then with high probability

$$
|H(\rho_A) - H^*(\mathrm{spec}(\phi_{\gamma_{A,1}}), \mathrm{spec}(\phi_{\gamma_{A,2}}))| = \mathcal{O}(\log(n)).
$$

Proof. The basic proof strategy will be that of Theorem 4.12: We study a slightly reduced problem on the approximated tensor network states $\tilde{\sigma}$ with approximate background states $\tilde{\phi}$, work out the entropies for $\tilde{\sigma}$ and $\tilde{\phi}$, and then argue that the closeness of the resulting entropies will continue to hold for the original tensor network state and background state, up to errors that we carefully keep track of.

First of all, we note that we can reduce to the tensor network state $\sigma$ on
the reduced graph $G'$, with error as in Eq. (4.23); in particular

$$
\mathbb{E} \| \mathrm{spec}_+(\rho_A) - \mathrm{spec}_+(\sigma_B) \|_1 = \mathcal{O}(n^{-\frac{\gamma}{2}}). \quad (4.46)
$$

Next we adapt the part of the proof of Theorem 4.12 where we modify
the state along the minimal cuts. In the proof of Theorem 4.12, we first ob-
served that if the (regularized) spectrum along each cut $\nu_{\phi_{\Gamma_{A,i}}}$ has uniformly
exponentially decaying tail probabilities, then for sufficiently large $C$, we can
slice off the tails with vanishing probability mass:

$$
\sum_{\lambda_{i,j} \notin I_n} \lambda_{i,j} = \nu_{\phi_{\Gamma_{A,i}}} ((-\infty, -C \log(n)) \cup (C \log(n), \infty)) = O\left(\frac{1}{n^4}\right).
$$
We also binned the eigenvalues of $\phi$ in the reduced spectrum:

$$q_{i,j} = d_{i,j} p_{i,j},$$

where $p_{i,j}$ are the bins and $d_{i,j}$ is the multiplicity of each bin. We then removed bins that were too small, leading to a state $\tilde{\phi}$. In the notation of the proof of Theorem 4.12 we choose $\alpha$ and $\beta$ such that $\alpha > 2$ and $\frac{5}{2} + \alpha < \beta \le \frac{1}{2}(\gamma - 1)$. By Eq. (4.26) this yields an error

$$\|\tilde{\phi} - \phi\|_1 = o(n^{-1}). \qquad (4.47)$$

Note that in the proof of Theorem 4.12, we performed one more approximation of removing the “middle” or “diagonal” part of the spectrum and obtained a state $\tilde{\phi}$; we will not need to do this here. Now, we recall that the binning of eigenvalues allows us to write $\tilde{\phi}$ as a superposition over maximally entangled states along each cut:

$$|\tilde{\phi}_{\gamma_{A,1}, \gamma_{A,2}}\rangle = \sum_{j,k} \sqrt{q_{1,j}q_{2,k}} |\Phi_{1,j}^{+}\rangle \otimes |\Phi_{2,k}^{+}\rangle,$$

and use this to decompose $\tilde{\phi}$ as

$$|\tilde{\phi}\rangle = \sum_{j,k} \sqrt{q_{1,j}q_{2,k}} |\psi^{(j,k)}\rangle,$$

where in $|\psi^{(j,k)}\rangle$, we have replaced $|\tilde{\phi}_{\gamma_{A,1}, \gamma_{A,2}}\rangle$ with the maximally entangled state $|\Phi_{1,j}^{+}\rangle \otimes |\Phi_{2,k}^{+}\rangle$. The states $|\psi^{(j,k)}\rangle$ are normalized background states on the graph $G'$. We let

$$|\phi^{(j,k)}\rangle = \sqrt{q_{1,j}q_{2,k}} |\psi^{(j,k)}\rangle$$

which are subnormalized states. At this point, the key idea of the argument is straightforward. We will consider the random tensor network states which have (a smoothed version) of the background states $|\psi^{(j,k)}\rangle$. These will have entropy close to $\min(\log(d_{1,j}), \log(d_{2,k}))$. Then, we will use Lemma 4.13 to argue that the entropy of $\rho_A$ is approximated up to $O(\log(n))$ terms by the convex combination $\sum_{j,k} q_{1,j}q_{2,k} \min(\log(d_{1,j}), \log(d_{2,k}))$, which we can relate to the desired result. To make this easy argument precise, we will need to take care of smoothing the background state appropriately and ensure that the relevant states are close to normalized.

We will now argue that we can smoothen the states $\phi^{(j,k)}$. We may assume without loss of generality that the states $\phi^{(j,k)}$ have nonnegative coefficients in the standard basis $|I\rangle = \prod_{e \in E'} |i_e i_e\rangle$ where $I = \{i_e\}_{e \in E'}$ runs over all possible basis elements over each edge, so we can write

$$|\phi^{(j,k)}\rangle = \sum_{I} \sqrt{\lambda_{I}^{(j,k)}}.$$

By the same argument as in Theorem 4.12 we find a state

$$\phi^{(j,k,\varepsilon)} = \sum_{I} \sqrt{\lambda_{I}^{(j,k,\varepsilon)}}$$
which is such that for any $\Delta_B \in C(B)$ not equal to $B$ or $V'_b \cup B$ and $\Delta_A = \Gamma_{A,1} \cup \Delta_B$

$$H_{\min}(\Delta_B \setminus B|B)_{\phi(j,k,\varepsilon)} \geq H_{\min}^{\varepsilon}(\Delta_A \setminus \Gamma_A|\Gamma_A)_{\phi},$$

while for $\Delta_B = B$ and $\Delta_B = B \cup V'_b$ we have

$$H_{\min}(B)_{\phi(j,k,\varepsilon)} \geq H_{\min}(B)_{\phi(j,k)} = \log \frac{1}{q_{2,k}p_{1,j}}$$

$$H_{\min}(\bar{B})_{\phi(j,k,\varepsilon)} \geq H_{\min}(\bar{B})_{\phi(j,k)} = \log \frac{1}{q_{1,j}p_{2,k}}.$$

Moreover, $\phi^{(j,k,\varepsilon)}$ is close to $\phi^{(j,k)}$ in the sense that

$$\|\phi^{(j,k,\varepsilon)} - \phi^{(j,k)}\|_1 = \mathcal{O}\left(\sqrt{\sum_I |\lambda^{(j,k,\varepsilon)} - \lambda^{(j,k)}|}\right) = \mathcal{O}(\sqrt{\varepsilon}).$$

By the remark after Lemma 4.11 we may assume that $\lambda^{(j,k,\varepsilon)} \leq \lambda^{(j,k)}$.

By a chain rule (e.g., Theorem 5.13 in [74], proven in [30]) for any $\Delta_B \in C(B)$ it holds that

$$H_2(\Delta_B)_{\phi^{(j,k,\varepsilon)}} \geq H_{\min}(\Delta_B \setminus B|B)_{\phi^{(j,k,\varepsilon)}} + H_{\min}(B)_{\phi^{(j,k,\varepsilon)}}.$$

We now let

$$|\phi^{\varepsilon}\rangle = \sum_{j,k} |\phi^{(j,k,\varepsilon)}\rangle.$$

Then, by Lemma C.4

$$\begin{aligned} \|\tilde{\phi} - \phi^{\varepsilon}\|_1 &\leq \sqrt{2 \sum_{j,k} \sum_I |\lambda_I^{(j,k)} - \lambda_I^{(j,k,\varepsilon)}|} = \mathcal{O}(\sqrt{n^{2\alpha} \log(n)^2 \varepsilon(n)}) \\ &= \mathcal{O}(n^{\alpha - \frac{1}{2}\gamma} \log(n)) = \mathcal{O}(n^{-3}). \end{aligned} \quad (4.48)$$

Therefore, if we let $\sigma_B^{\varepsilon}$ be the random tensor network state with background state $\phi^{\varepsilon}$

$$\mathbb{E}\|\sigma - \sigma^{\varepsilon}\|_1 \leq \|\phi - \tilde{\phi}\|_1 + \|\tilde{\phi} - \phi^{\varepsilon}\|_1 = o\left(\frac{1}{n}\right). \quad (4.49)$$

Finally, let

$$|\phi^{(j,k,\varepsilon)}\rangle = \sqrt{q_{1,j}q_{2,k}}|\psi^{(j,k,\varepsilon)}\rangle$$

By Eq. (C.2) in the remark after the proof of Lemma 4.11 we have

$$\begin{aligned} \left|\mathrm{tr}[\psi^{(j,k,\varepsilon)}] - 1\right| &= \frac{1}{q_{1,j}q_{2,k}} \left|\mathrm{tr}[\phi^{(j,k,\varepsilon)}] - \mathrm{tr}[\phi^{(j,k)}]\right| \\ &= \mathcal{O}\left(\frac{\varepsilon}{q_{1,j}q_{2,k}}\right) = \mathcal{O}(n^{2\beta-\gamma}) = \mathcal{O}(n^{-1}). \end{aligned}$$

using that $q_{1,j}, q_{2,k} > n^{-\beta}$ and $\beta \leq \frac{1}{2}(\gamma - 1)$. We denote the random tensor network states with background states $\psi^{(j,k,\varepsilon)}$ by $\sigma^{(j,k)}$, and the random tensor network states with background states $\sum_j \sqrt{q_{1,j}} \psi^{(j,k,\varepsilon)}$ by $\sigma^{(k)}$.
We introduce the event $N$, which entails that $\rho, \sigma$ and $\sigma^{(j,k)}$ for all $j, k$ are close to normalized, that is

$$|\mathrm{tr}[\rho] - 1| \le \frac{1}{n^2} \quad \text{and} \quad |\mathrm{tr}[\sigma] - 1| \le \frac{1}{n^2} \quad \text{and} \quad \left|\mathrm{tr}[\sigma^{(j,k)}] - \mathrm{tr}[\psi^{(j,k,\varepsilon)}]\right| \le \frac{1}{n^2}.$$

By assumption, for each edge $e = (xy)$ we have $\mathrm{tr}[\phi_{e,x}^2] \le 2^{-\Omega(n)}$. Moreover, using that $q_{1,j}, q_{2,k} \ge n^{-\beta}$

$$\mathrm{tr}[(\phi_{\Delta}^{(j,k,\varepsilon)})^2] \le q_{1,j}^{-2} q_{2,k}^{-2} \mathrm{tr}[\phi_{\Delta}^2] = \mathcal{O}(\mathrm{poly}(n)2^{-\Omega(n)})$$

and therefore the quantity $\eta$ in Lemma 2.1 is $\mathcal{O}(2^{-\Omega(n)})$ for $\rho, \sigma$ and $\sigma^{(j,k)}$. So, by Lemma 2.1 and the union bound the event $N$ has probability

$$\begin{aligned} p_N \ge 1 - & \left( \mathrm{Pr} \left( |\mathrm{tr}[\rho] - 1| \ge \frac{1}{n^2} \right) + \mathrm{Pr} \left( |\mathrm{tr}[\sigma] - 1| \ge \frac{1}{n^2} \right) \right. \\ & \left. + \sum_{j,k} \mathrm{Pr} \left( \left| \mathrm{tr}[\sigma^{(j,k)}] - \mathrm{tr}[\psi^{(j,k,\varepsilon)}] \right| \ge \frac{1}{n^2} \right) \right) \\ = 1 - \mathcal{O}(\mathrm{poly}(n)2^{-\Omega(n)}) = 1 - \mathcal{O}(2^{-\Omega(n)}). \end{aligned}$$

We denote by $\mathbb{E}_N$ the expectation value over the random tensors conditioned on this event.

We now use Lemma 4.13 to approximate the entropy of $\sigma_B^\varepsilon$ conditioned on $N$:

$$\left| H(\sigma_B^\varepsilon) - \sum_k q_{2,k} H(\sigma_B^{(k)}) \right| = \mathcal{O}(\log(n)).$$

Because $\sigma^{(k)}$ is pure, we have $H(\sigma_B^{(k)}) = H(\sigma_B^{(k)}).$ We apply Lemma 4.13 again, this time to the decomposition of $\sigma_B^{(k)}$ to see:

$$\left| H(\sigma_B^{(k)}) - \sum_j q_{1,j} H(\sigma_B^{(j,k)}) \right| = \mathcal{O}(\log(n)).$$

Thus,

$$\left| H(\sigma_B^\varepsilon) - \sum_{j,k} q_{1,j} q_{2,k} H(\sigma_B^{(j,k)}) \right| = \mathcal{O}(\log(n)).$$

Since $\mathrm{rank}(\sigma_B^{(j,k)}) \le \min\{d_{1,j}, d_{2,k}\}$ (and taking into account the normalization of $\sigma^{(j,k)})$

$$H(\sigma_B^{(j,k)}) \le \mathrm{tr}[\sigma^{(j,k)}] \min(\log d_{1,j}, \log d_{2,k}) - \mathrm{tr}[\sigma^{(j,k)}] \log \mathrm{tr}[\sigma^{(j,k)}].$$

Conditioned on $N$ and using $|\mathrm{tr}[\psi^{(j,k,\varepsilon)}] - 1| = \mathcal{O}(n^{-1})$, this implies

$$H(\sigma_B^{(j,k)}) \le \min(\log d_{1,j}, \log d_{2,k}) + \mathcal{O}(1).$$
For a lower bound, we use that

$$
\begin{align*}
\mathbb{E}_N H(\sigma_B^{(j,k)}) &\ge \mathbb{E}_N \operatorname{tr}[\sigma^{(j,k)}] \left(-\log \operatorname{tr}[(\sigma_B^{(j,k)})^2] + \log \operatorname{tr}[\sigma_B^{(j,k)}]\right) \\
&\ge \left(\operatorname{tr}[\psi^{(j,k,\varepsilon)}] - \frac{1}{n^2}\right) \left(-\log \mathbb{E}_N \operatorname{tr}[(\sigma_B^{(j,k)})^2] + \log \left(\operatorname{tr}[\psi^{(j,k,\varepsilon)}] - \frac{1}{n^2}\right)\right) \\
&\ge -\log \mathbb{E}_N \operatorname{tr}[(\sigma_B^{(j,k)})^2] - \mathcal{O}(1)
\end{align*}
$$

where in the first and second inequality we have used Jensen's inequality, and
in the second and third inequality we have used tr[$\sigma_B^{(j,k)}$] $\ge$ tr[$\psi^{(j,k,\epsilon)}$] $-$ $\frac{1}{n^2}$ $\ge$
$1 - \mathcal{O}(\frac{1}{n})$. We use the replica trick to estimate

$$
\mathbb{E} \operatorname{tr} [ (\sigma_B^{(j,k)})^2 ] = \sum_{\Delta_B \in C(B)} \operatorname{tr} [ (\psi_{\Delta_B}^{(j,k,\epsilon)})^2 ]
$$

In this expression, we have contributions from $\Delta_B = B$ and $\Delta_B = B \cup V'_b$,
which yield contributions

$$
\begin{align*}
\operatorname{tr}[(\psi_B^{(j,k,\epsilon)})^2] &\le d_{1,j}^{-1} \\
\operatorname{tr}[(\psi_B^{(j,k,\epsilon)})^2] &\le d_{2,k}^{-1}.
\end{align*}
$$

For any other cut $\Delta_B$, we have a contribution at most

$$
\begin{align*}
\operatorname{tr}[(\psi_{\Delta_B}^{(j,k,\epsilon)})^2] &= q_{1,j}^{-2} q_{2,k}^{-2} \operatorname{tr}[\phi^{(j,k,\epsilon)}] 2^{-H_{\min}(B)_{\phi^{(j,k,\epsilon)}} - H_{\min}^{\epsilon}(\Delta_A \setminus \Gamma_A | \Gamma_A)_{\phi}} \\
&\le q_{1,j}^{-1} q_{2,k}^{-1} 2^{-H_{\min}(B)_{\phi^{(j,k,\epsilon)}}} \le d_{1,j}^{-1}
\end{align*}
$$

using that tr[$\phi^{(j,k,\epsilon)}$] $\le$ q₁,jq₂,k and 2⁻¹Hmin(B)φ(j,k,ε) $\le$ q₂,kp₁,j = q₁,jq₂,kd⁻¹₁,j.
Therefore

$$
\begin{align*}
\mathbb{E}_N \operatorname{tr} [ (\sigma_B^{(j,k,\epsilon)})^2 ] &\le p_N^{-1} \mathbb{E} \operatorname{tr} [ (\sigma_B^{(j,k,\epsilon)})^2 ] \\
&= d_{1,j}^{-1} + d_{2,k}^{-1} + \mathcal{O}(d_{1,j}^{-1}).
\end{align*}
$$

so

$$
\begin{align*}
-\log \mathbb{E}_N \operatorname{tr} [ (\sigma_B^{(j,k,\epsilon)})^2 ] &\ge -\log \left( \max \left( d_{1,j}^{-1}, d_{2,k}^{-1} \right) (2 + \mathcal{O}(1)) \right) \\
&\ge \min(\log d_{1,j}, \log d_{2,k}) - \mathcal{O}(1).
\end{align*}
$$

We find that

$$
\mathbb{E}_N \left| H(\sigma_B^{(j,k)}) - \min(\log d_{1,j}, \log d_{2,k}) \right| = \mathcal{O}(1)
$$

and hence

$$
\mathbb{E}_N \left| \sum_{j,k} q_{1,j} q_{2,k} H(\sigma_B^{(j,k)}) - \sum_{j,k} q_{1,j} q_{2,k} \min(\log d_{1,j}, \log d_{2,k}) \right| = \mathcal{O}(1). \quad (4.50)
$$
We collect the various estimates we have found:

$$ \mathbb{E}_N \| \text{spec}_+(\rho_A) - \text{spec}_+(\sigma_B) \|_1 \le p_N^{-1} \mathbb{E} \| \text{spec}_+(\rho_A) - \text{spec}_+(\sigma_B) \|_1 = o\left(\frac{1}{n}\right) $$

$$ \mathbb{E}_N \|\sigma_B - \sigma_B^\varepsilon\|_1 \le p_N^{-1} \mathbb{E} \|\sigma_B - \sigma_B^\varepsilon\|_1 = o\left(\frac{1}{n}\right) $$

$$ \mathbb{E}_N \left| \sum_{j,k} q_{1,j} q_{2,k} H(\sigma_B^{(j,k)}) - \sum_{j,k} q_{1,j} q_{2,k} \min(\log d_{1,j}, \log d_{2,k}) \right| = \mathcal{O}(1). $$

Let $M$ be the event that $N$ holds and moreover

$$ \|\text{spec}_+(\rho_A) - \text{spec}_+(\sigma_B)\|_1 \le \frac{1}{n} $$

$$ \|\sigma_B - \sigma_B^\varepsilon\|_1 \le \frac{1}{n} $$

$$ \left| \sum_{j,k} q_{1,j} q_{2,k} H(\sigma_B^{(j,k)}) - \sum_{j,k} q_{1,j} q_{2,k} \min(\log d_{1,j}, \log d_{2,k}) \right| \le \log(n). $$

By Markov's inequality and the union bound, the probability that $M$ holds goes to one as $n$ goes to infinity. Moreover, if $M$ holds, it is easy to verify that by the Fannes–Audenaert inequality Eq. (4.44) and the fact that $\rho$, $\sigma$ and $\sigma^\varepsilon$ are close to normalized

$$ \left| H\left(\frac{\rho_A}{\text{tr}[\rho]}\right) - H\left(\sigma_B^\varepsilon\right) \right| = \mathcal{O}(1). $$

Also, if $M$ holds, by Eq. (4.50)

$$ \left| H(\sigma_B^\varepsilon) - \sum_{j,k} q_{1,j} q_{2,k} \min(\log d_{1,j}, \log d_{2,k}) \right| = \left| \sum_{j,k} q_{1,j} q_{2,k} H(\sigma_B^{(j,k)}) - \sum_{j,k} q_{1,j} q_{2,k} \min(\log d_{1,j}, \log d_{2,k}) \right| + \mathcal{O}(\log(n)) = \mathcal{O}(\log(n)) $$

so we conclude that

$$ \left| H\left(\frac{\rho_A}{\text{tr}[\rho]}\right) - \sum_{j,k} q_{1,j} q_{2,k} \min(\log d_{1,j}, \log d_{2,k}) \right| = \mathcal{O}(\log(n)). $$

Since $q_{i,j} = d_{i,j} p_{i,j} \ge n^{-\beta}$ from the binning procedure, we have the simple observation

$$ \frac{1}{n^\beta p_{i,j}} \le \frac{q_{i,j}}{p_{i,j}} = d_{i,j} \le \frac{1}{p_{i,j}}, $$
and hence $|\min(\log d_{1,j}, \log d_{2,k}) - \log(\min(\frac{1}{p_{1,k}}, \frac{1}{p_{2,j}}))| = \mathcal{O}(\log(n))$. We conclude that

$$ \left| H \left( \frac{\rho_A}{\text{tr}[\rho]} \right) - \sum_{j,k} q_{1,k} q_{2,j} \log \left( \min \left( \frac{1}{p_{1,k}}, \frac{1}{p_{2,j}} \right) \right) \right| = \mathcal{O}(\log(n)). $$

Finally, we need to relate the result back to the original background state. In the above approximation to $H(\tilde{\sigma}_B)$, we see that

$$ \sum_{j,k} q_{1,k} q_{2,j} \log \left( \min \left( \frac{1}{p_{1,k}}, \frac{1}{p_{2,j}} \right) \right) = H^*(\text{spec}(\tilde{\phi}_{\gamma_{A,1}}), \text{spec}(\tilde{\phi}_{\gamma_{A,2}})). $$

Then by Eqs. (4.47) and (4.45), this will converge to the appropriate quantity on the non-approximated background state:

$$ |H^*(\text{spec}(\tilde{\phi}_{\gamma_{A,1}}), \text{spec}(\tilde{\phi}_{\gamma_{A,2}})) - H^*(\text{spec}(\phi_{\gamma_{A,1}}), \text{spec}(\phi_{\gamma_{A,2}}))| \to 0, $$

proving the desired result. □

## Acknowledgements

NC is supported in part by the Department of Energy via the GeoFlow consortium (QuantISED Award DE-SC0019380). CL acknowledges support from the projects ESQuisses (ANR-20-CE47-0014-01), STARS (ANR-20-CE40-0008), Qtraj (ANR-20-CE40-0024-01), and Random Tensors (ANR-11-LABX-0040) of the French National Research Agency (ANR). GP is supported by the UC Berkeley Physics Department, the Simons Foundation through the “It from Qubit” program, the Department of Energy via the GeoFlow consortium (QuantISED Award DE-SC0019380), and AFOSR award FA9550-22-1-0098. He also acknowledges support from an IBM Einstein Fellowship at the Institute for Advanced Study. MW acknowledges support by the NWO through Grant OCENW.KLEIN.267, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972, by the BMBF through project Quantum Methods and Benchmarks for Resource Allocation (QuBRA), and by European Research Council (ERC), funded by the European Union. Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

**Funding** Open access funding provided by Royal Library, Copenhagen University Library.
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# A. Euclidean Gravity Path Integrals and Entanglement Spectra

In this appendix, we give a heuristic description of certain Euclidean gravity path integrals in holography whose descriptions are in close analogy with random tensor network models. This section serves as a motivation for the random tensor network models we study, but is not needed to understand the random tensor network results. In “Appendix A.1,” we review the replica trick in quantum field theory and the role of Euclidean path integrals. We review an application of such tools in “Appendix A.2,” and we discuss the problem of studying the entanglement entropy near a phase transition between two minimal surfaces, which agrees with results in Sect. 4. Then, in “Appendix A.3,” we discuss a simplified model of quantum gravity which is in very close correspondence to a random tensor network model with link states with bounded spectral variation, as in Sect. 3.

## A.1. The Replica Trick and Euclidean Path Integrals

In order to compute entropies in quantum field theory, one often uses a version of the replica trick to compute the Rényi entropies, which can then be analytically continued to deduce von Neumann entropies. We consider a pure quantum field theory state $|\rho\rangle$ on a space $M$, which is prepared by a Euclidean path integral on $M \times (-\infty, 0]$. Correspondingly, $\langle\rho|$ is prepared by the time-reflected path integral on $M \times [0, \infty)$. Let $A$ be a subregion of $M$. The reduced density matrix on $A$ is given by taking $|\rho\rangle$ and $\langle\rho|$, then integrating over the field configurations on the complement of $A$ (the equivalent of the partial trace for field theories), as shown in Fig. 6a. Analogous to Eq. (2.8), we may now compute $\text{tr}[\rho_A^k]$ by taking $k$ copies of this path integral, and gluing the boundaries at the $A$ system cyclically, then integrating over the field configurations at each boundary. This operation is manifestly invariant under cyclic permutations, a symmetry called replica symmetry. This is illustrated in Fig. 6b. We conclude that $\text{tr}[\rho_A^k]$ is computed by a path integral $Z_{A,k}$ on a manifold $M_{A,k}$, allowing us to compute $H_k(\rho_A)$. The space $M_{A,k}$ is a $k$-fold cover of $M \times \mathbb{R}$, branching at the boundary $\partial A$ of the subregion $A$.
<figure><img src="image_24.png" /><figcaption>FIGURE 6. The path integral replica trick to compute tr[ρ̃<sub>A</sub><sup>k</sup>]

*FIGURE 6.*</figcaption></figure>

Of course, the path integral $Z_{A,k}$ is formally infinite. One “normalizes” the path integral by normalizing by $Z_1$ which can be thought of as tr[$\rho$]. We have the expression for the $k$th Rényi entropy:

$$H_k(\rho_A) = \log \frac{Z_{A,k}}{Z_1^k}. \qquad (A.1)$$

Note that this is a slightly different normalization convention (the denominator is $Z_1^k$ instead of $Z_1$) to match the standard convention in the quantum gravity literature. To get a finite result, one has to impose a UV cutoff of size $\epsilon$, an aspect we will ignore in our discussion. By employing such a cutoff, we will pretend that the relevant Hilbert spaces are finite-dimensional Hilbert spaces that factorize with respect to spatial decompositions of $M$.

What happens if the quantum field theory is a holographic CFT? In this case, we have a correspondence between the path integral of the CFT on the one hand, and a bulk quantum gravity path integral on the other hand, which, for large effective central charge, we may approximate by its semiclassical saddle point geometry. This can be used to derive the Ryu-Takayanagi formula [45], by finding a bulk manifold $B_{A,k}$ which has $M_{A,k}$ as its boundary. In other words, one obtains the gravitational dual by setting $M_{A,k}$ as a boundary condition, then allowing the system to evolve according to the gravitational theory.

The saddle point solution will be such that the bulk copies are glued along a cyclic permutation in an area adjacent to $A$ and along the identity permutation in an area adjacent to $\bar{A}$, the boundary complement of $A$. The boundary between these two regions is a surface $\gamma_{A,k}$ anchored at the boundary at $\partial A$. After orbifolding with respect to the replica symmetry, this leads to a bulk spacetime which has $M$ as its boundary and a “cosmic brane” at the surface $\gamma_{A,k}$. The orbifold procedure gives rise to a conical deficit, corresponding to the fixed points under the action of the replica symmetry. Comparing
the action of $B_{A,k}$ and $B_1$, we find that in the computation of Eq. (A.1), only the conical deficit at $\gamma_{A,k}$ does not cancel, yielding

$$H_k(\rho_A) \approx \frac{|\gamma_{A,k}|}{4G_N}. \qquad (A.2)$$

where $|\gamma_{A,k}|$ is the area of the brane in the saddle point solution. This can be analytically continued to non-integer values, and in particular, continuation to $k = 1$ yields the RT formula:

$$H(\rho_A) \approx \frac{|\gamma_A|}{4G_N}. \qquad (A.3)$$

where $\gamma_A$ is the surface of minimal area out of all bulk surface homologous to A and anchored at $\partial A$.

The spectrum of $\rho_A$ can be recovered from the Rényi entropies. In Eq. (A.2), we see that $|\gamma_{A,k}|$ depends on $k$, but is otherwise fixed as we let $G_N$ go to zero in the classical limit. It was argued in [14] that this behavior implies a spectrum that is flat at leading order:

$$H_{\min}^{\epsilon}(\rho_A) = H(\rho_A) - \mathcal{O}(G_N^{-\frac{1}{2}}) = H(\rho_A) - \mathcal{O}(\sqrt{H(\rho_A)}) \\ H_{\max}^{\epsilon}(\rho_A) = H(\rho_A) + \mathcal{O}(G_N^{-\frac{1}{2}}) = H(\rho_A) + \mathcal{O}(\sqrt{H(\rho_A)})$$

This is precisely the link state regime we investigated in Sect. 4.4, and shows that the large $c$ limit in a holographic CFT is similar to considering the many-copy limit in quantum information theory.

**A.1.1. Fixed-Area States.** A useful variation on the derivation of the RT formula is to consider fixed-area states [27]. We consider the “area operator” $\hat{\gamma}_A$ for a subsystem A, which measures the area of a minimal surface. The operator $\hat{\gamma}_A$ actually has fluctuations when we consider a state with a semiclassical gravity dual, and we may write

$$|\rho\rangle = \int d\alpha |\psi_{\alpha}\rangle$$

where $\psi_\alpha$ is an eigenvector of $\hat{A}$ with eigenvalue $\alpha$. The state $|\psi_\alpha\rangle$ is a fixed-area state: While it is not itself a physical state, they form a basis with which to construct physical states. These fixed-area states can be thought of as prepared by a bulk path integral where we have restricted to bulk geometries for which $|\gamma_A| = \alpha$. The same derivation as above now leads to

$$H_k(\rho_A) \approx \frac{\alpha}{4G_N},$$

since the area of the minimal surface is fixed to be $\alpha$. Thus, for fixed-area states, all Rényi entropies are (to good approximation) equal, which implies that the state has flat entanglement spectrum. This corresponds to a random tensor network state with maximally entangled link states.
## A.2. States at a Minimal Surface Phase Transition

Consider, again, a holographic CFT state $\rho_A$ on a boundary subregion A. In the case where there is a unique minimal surface, the previous subsection showed how the replica trick leads to the RT formula.

We would now like to investigate the entanglement spectrum of $\rho_A$ when there are two RT surfaces for A that have area of the same order of magnitude. Let us denote the two competing minimal surfaces in the bulk by $\gamma_{A,1}$ and $\gamma_{A,2}$. In [54], it was shown how the entanglement entropy should behave at this phase transition between the two minimal surfaces. A similar computation was performed in [6] for the setting with two competing minimal surfaces and bulk matter. We will briefly sketch their argument, referring the interested reader to [6] for more details.

### A.2.1. Fixed-Area States with Two Minimal Surfaces.
We begin by considering bipartite fixed-area states $\rho_{A\bar{A}}$ prepared by a Euclidean path integral, in which we have fixed the size of the two competing surfaces $\gamma_{A,1}$ and $\gamma_{A,2}$. In this case, saddle points of the path integral have to satisfy the equations of motion everywhere except at the surfaces $\gamma_{A,i}$, where there could be conical singularities. The two surfaces divide the bulk into 3 regions: $a_1$, $a_2$, and $a_3$.

The saddle points are states with smooth geometries in the regions where the copies of regions $a_i$ are glued to each other—the $k$ copies of region $a_1$ are glued cyclically, while the copies of region $a_3$ are glued along the identity permutation. On the middle region $a_2$, we are free to glue along an arbitrary permutation $\pi$, giving solutions that can break the replica symmetry. However, it is often the case that the dominant solutions of the path integral are those that respect the replica symmetry, and we assume this to be the case for all solutions we consider.

Let us denote a saddle point solution of the path integral with permutation $\pi$ by $B_{A,\pi}$, and let us write $\phi_i$ for the conical singularity angle at $\gamma_{A,i}$. It turns out that these saddle points lead to an action of the form

$$I(B_{A,\pi}) = kI_{\text{away}}(B_{A,\pi}) + (k\phi_1 - 2\pi|C(\pi)|)\frac{|\gamma_{A,1}|}{8\pi G_N} + (k\phi_2 - 2\pi|C(\tau^{-1}\pi)|)\frac{|\gamma_{A,2}|}{8\pi G_N},$$

where $I_{\text{away}}$ is the action away from the surfaces, $|\gamma_{A,i}|$ is the area of the surface $\gamma_{A,i}$, and we recall that $|C(\pi)|$ is the number of cycles of $\pi$, and $\tau$ is the full cycle (12...k). In particular, for $k=1$, we have

$$I(B_{A,\pi}) = I_{\text{away}}(B_{A,\pi}) + (\phi_2 - 2\pi)\frac{|\gamma_{A,1}|}{8\pi G_N} + (\phi_2 - 2\pi)\frac{|\gamma_{A,2}|}{8\pi G_N},$$
so when we look at the normalized path integral and sum over all permutations

$$ \frac{Z_{A,k}}{(Z_{A,1})^k} \approx \sum_{\pi \in S_k} e^{(|C(\pi)|-k) \frac{|\gamma_{A,1}|}{4G_N} + (|C(\tau^{-1}\pi)|-k) \frac{|\gamma_{A,2}|}{4G_N}} \\ = \sum_{\pi \in S_k} e^{d(\mathrm{id}, \pi) \frac{|\gamma_{A,1}|}{4G_N} + d(\pi, \tau) \frac{|\gamma_{A,2}|}{4G_N}}. $$

Note that the areas $\gamma_{A,i}$ are of the same order of magnitude, and are divergent. As a result, only the permutations for which $d(\mathrm{id}, \pi) + d(\pi, \tau)$ is minimal will contribute. In other words, only the configurations where $\pi \in NC(k)$ contribute, as all other permutations are suppressed by at least a factor of the area of $\gamma_{A,i}$ in the action. We conclude that

$$ \frac{Z_{A,k}}{(Z_{A,1})^k} \approx \sum_{\pi \in NC(k)} e^{d(\mathrm{id}, \pi) \frac{|\gamma_{A,1}|}{4G_N} + d(\pi, \tau) \frac{|\gamma_{A,2}|}{4G_N}}. \quad (A.4) $$

This computation is in one-to-one correspondence with the computation of the moments for a subsystem of a single random tensor, as observed in [60]. It also corresponds more generally to a random tensor network computation with two minimal cuts and maximally entangled link states, as is clear from the computations in Sects. 2 and 3. This is in agreement with the claim that random tensor network states are a model for fixed-area states. One can also add bulk matter in this path integral computation, which will again be in correspondence to a similar computation in a random tensor network [6] with a background state, as in Sect. 2.3. From the moment computation in Eq. (A.4) and applying the results for the entanglement of a single random tensor, we observe that for two fixed surfaces of exactly equal size, the (appropriately scaled) entanglement spectrum is a Marchenko-Pastur distribution, giving an $O(1)$ correction to the entanglement entropy, agreeing with the gravitational replica trick computation in [54].

**A.2.2. General States at the Minimal Surface Phase Transition.** We now relax the fixed-area restriction, and study similar calculations performed in [27], [54], and [6]. Denote by $Z_{A,k}(\alpha_1, \alpha_2)$ the path integral where we have fixed the areas of $\gamma_{A,i}$ to be $\alpha_i$. Then, following Section 2.3 in [27], the full path integral is given by

$$ Z_{A,k} = \int d\alpha_1 d\alpha_2 Z_k(\alpha_1, \alpha_2). $$

Again, we consider the semiclassical limit, so we take our saddle point approximation of $Z_k(\alpha_1, \alpha_2)$ in Eq. (A.4), and we also take a saddle point approximation for the integral over $\alpha_1$ and $\alpha_2$. This saddle point will be at the values for $\alpha_i$ where the deficit angles are $\phi_i = \frac{2\pi}{n}$ (since then the saddle point geometry is smooth), which leads to

$$ \frac{Z_{A,k}}{(Z_{A,1})^k} \approx \sum_{\pi \in NC(k)} e^{(|C(\pi)|-k) \frac{|\gamma_{A,1}^{(k)}|}{4G_N} + (|C(\tau^{-1}\pi)|-k) \frac{|\gamma_{A,2}^{(k)}|}{4G_N}} \quad (A.5) $$
where $\gamma_{A,i}^{(k)}$ are now minimal surfaces, with a dependence on $k$. Analytic continuation to $k=1$ yields the usual surface prescription. In particular, if there are two surfaces that are of almost equal area, the contribution of the larger term is exponentially suppressed for any $O(1)$ or larger difference in areas.

To zoom in on the region where the two surfaces are nearly equal, we write the state as a superposition of fixed-area states, as in Section 3 of [54]. We discretize the area size $\alpha_1$ and $\alpha_2$ over poly($\frac{1}{G_N}$) values and approximate the full holographic pure state with boundary regions $A$ and $B = A^c$ as a (finite) sum

$$|\psi\rangle_{AB} = \sum_{\alpha_1, \alpha_2} \sqrt{p(\alpha_1, \alpha_2)} |\psi_{\alpha_1, \alpha_2}\rangle$$

where $|\psi_{\alpha_1, \alpha_2}\rangle$ is the state where the areas are fixed as $|\gamma_{A,i}| = \alpha_i$, and $p$ is a probability distribution over the possible areas. Then, a straightforward calculation of the reduced density matrix $\rho_A$ yields a state of the form:

$$\rho_A = \sum_{\alpha_1, \alpha_2} p(\alpha_1, \alpha_2) \rho_{A, \alpha_1, \alpha_2} + \sum_{\substack{\alpha_1 \neq \alpha'_1, \alpha_2 \neq \alpha'_2 \\ (|\psi_{\alpha_1, \alpha_2}\rangle\langle\psi_{\alpha'_1, \alpha'_2}|)}} \sqrt{p(\alpha_1, \alpha_2) p(\alpha'_1, \alpha'_2)} \text{tr}_B \\ = \sum_{\alpha_1, \alpha_2} p(\alpha_1, \alpha_2) \rho_{A, \alpha_1, \alpha_2} + OD_A,$$

where $OD_A$ are the off-diagonal elements of $\rho_A$. One can argue that the states $\rho_{A,\alpha_1,\alpha_2}$ are all mutually orthogonal by entanglement wedge reconstruction—the area operator can be reconstructed on $A$, and hence, each $\rho_{A,\alpha_1,\alpha_2}$ is perfectly distinguishable from each other by measuring the area operator. Then, the entropy of the diagonal part of the state is easily computed as

$$H\left(\sum_{\alpha_1, \alpha_2} p(\alpha_1, \alpha_2) \rho_{A, \alpha_1, \alpha_2}\right) = \sum_{\alpha_1, \alpha_2} p(\alpha_1, \alpha_2) H(\rho_{A, \alpha_1, \alpha_2}) \\ - \sum_{\alpha_1, \alpha_2} p(\alpha_1, \alpha_2) \log p(\alpha_1, \alpha_2). \quad (A.6)$$

The second term is the so-called entropy of mixing, and it is a standard argument that this term is suppressed relative to the first term [54] as $O(\ln G_N)$ or smaller. The entropies appearing in the first term can be computed using the methods in the previous subsection, for which one finds that $H(\rho_{A,\alpha_1,\alpha_2}) = \frac{\min\{\alpha_1, \alpha_2\}}{4G_N}$.

Returning to the off-diagonal terms $OD_A$, [54] argued that such terms should be subleading in the analytic continuation due to the relevant surfaces breaking replica symmetry. At the same time, [6] argued that such terms should be subleading due to reasons similar to those for the orthogonality of the diagonal elements: Complementary entanglement wedge reconstruction implies one may reconstruct the bulk area operator on $B$, and hence, such states are perfectly distinguishable on $B$. Therefore, the partial trace over $B$ vanishes for $\alpha_1 \neq \alpha'_1, \alpha_2 \neq \alpha'_2$.
One reaches the conclusion:

$$H(\rho_A) = \sum_{\alpha_1, \alpha_2} p(\alpha_1, \alpha_2) \frac{\min\{\alpha_1, \alpha_2\}}{4G_N} + \mathcal{O}(\ln G_N), \quad (A.7)$$

In this computation, the $\mathcal{O}(1)$ corrections due to the Marchenko-Pastur distribution along each pair of minimal cuts of equal size (or equivalently, the degeneracy in the contributions to the saddle point approximation) is irrelevant, as the entropy of mixing already leads to $\mathcal{O}(\ln G_N)$ deviations.

Our results in Sect. 4 can be seen as a rigorous version of the above result for random tensor networks.

### A.3. Replica Wormholes and JT Gravity

One of the most basic holographic models of quantum gravity is JT gravity, a 1+1-dimensional model of gravity; see [67] for a review. JT gravity is also a useful model for the near-horizon dynamics of extremal black holes in any dimension. In this case, the dual theory should be 0+1-dimensional. In other words, it should be regular quantum mechanics rather than a quantum field theory. Indeed, in [68], it was shown that JT gravity theory is dual to a random matrix model, where the Hamiltonian is a random self-adjoint matrix according to some distribution, providing another strong connection between quantum gravity and random matrix theory. It also appears that such gravitational systems may be dual to an ensemble of boundary theories [16], rather than a single one. Whether this is fundamental, a special feature of 1+1-dimensional models, or due to averaging over microscopic features of the gravity theory, is a line of active research [69].

We now sketch a variation on a calculation in [60], providing a proof of principle that the free probability techniques used in Sect. 3 provide an elegant framework to understand such results. We refer the interested reader to [60] for more in-depth motivation and detailed computations.

We consider JT gravity with an end of the world (EOW) brane containing a large number $n$ of internal states. This model has action

$$I = I_{JT} + \mu \int_{\text{brane}} \mathrm{d}s,$$

where the action of a manifold $M$ with metric $g$, induced boundary metric $h$, (trace of) extrinsic curvature $K$, and dilaton $\phi$ is given by

$$I_{JT}[M, g] = -\frac{S_0}{2\pi} \left[ \frac{1}{2} \int_M \sqrt{g} R + \int_{\partial M} \sqrt{h} K \right] \\ - \left[ \frac{1}{2} \int_M \sqrt{g} \phi(R+2) + \int_{\partial M} \sqrt{h} \phi K \right].$$

The details of this action are not very important for us; we just note that we will take the $S_0$ parameter to be large, and that this suppresses contributions where the manifold $M$ has genus $\gamma > 0$ in the Euclidean path integral.
<figure><img src="image_25.png" /><figcaption>FIGURE 7. Path integrals and the replica trick for JT gravity. See [60] for a detailed explanation of the diagrammatic notation</figcaption></figure>

Such systems are of interest when studying a simple version of an evaporating black hole. Let

$$|\psi\rangle = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} |\psi_{B,i}\rangle |i_{\text{Rad}}\rangle$$

where $|\psi_{B,i}\rangle$ is the state of the black hole with the EOW brane in state $i$, and $|i_{\text{Rad}}\rangle$ is a reference state, which can be thought of as the radiation system. Notice that the entanglement spectrum of this state is flat. We generalize this to

$$|\psi\rangle = \sum_{i=1}^{n} \sqrt{p_i} |\psi_{B,i}\rangle |i_{\text{Rad}}\rangle,$$

where the entanglement between the black hole and the radiation has some non-trivial spectrum, which we will assume to be close to uniform, so that $p_i/n$ is bounded by a constant for all $i$. In other words, we assume this distribution satisfies the bounded spectral variation assumption from Sect. 3. We let

$$m_k = \sum_{i=1}^{n} n^{k-1} p_i^k$$

be the (appropriately scaled) moments of the entanglement spectrum of the EOW brane. Moreover, we write the path integral on a disk geometry with $k$ boundary components and $k$ EOW branes as $e^{-S_0} Z_k$. Then following the arguments of [60], one can compute the $k$th moment of the radiation system for large $n$ and large $e^{S_0}$ (large $n$ enforces a planar limit with only non-crossing partitions, while large $e^{S_0}$ ensures that only genus $\gamma = 0$ geometries contribute),
as illustrated in Fig. 7b. The contributions of path integral configurations connecting different replicas are called replica wormholes. This diagrammatic computation shows that

$$ \mathrm{tr}[\psi_{\mathrm{Rad}}^k] = \sum_{\pi \in NC(k)} m_{\pi} \frac{Z_{\pi^{-1}\tau}}{Z_1^k} n^{-d(\pi, \mathrm{id})} e^{-S_0 d(\pi, \tau)}. \quad (A.8) $$

In this expression, recall that $Z_σ = \prod_{l \in C(σ)} Z_l$, where $C(σ)$ is the cycle type of $σ$, and $l \in C(σ)$ are the lengths of the cycles of $σ$. This expression implies that if $n \gg e^{S_0}$, the dominant contribution in Eq. (A.8) is given by $π = τ$. On the other hand, if $n \ll e^{S_0}$, the dominant contribution in Eq. (A.8) is given by $π = \mathrm{id}$. This corresponds to the situation where there is a unique minimal surface (more precisely, a unique quantum extremal surface). We are interested in the regime at the phase transition, which is analogous to the Page time of an evaporating black hole, so we assume $ne^{-S_0} \to 1$. The coefficients $m_σ$ correspond to the weight of the $σ$ configuration, as determined by the number and length of the cycles in $σ$, and the probability distribution of eigenstates $p_i$. In the case of the flat entanglement spectrum, this number equals the number of closed loops between the connected components. This will also be the case for the non-trivial entanglement spectrum, but each loop will have a different weight that depend on the $p_i$'s.

The $m_k$ are the (scaled) moments of a probability distribution. While the explicit expression itself is not important for our purposes, the $Z_l$ can be written as the $l$th moments of a probability distribution [60]. Hence, Eq. (A.8) is a product of moments, summed over all non-crossing partitions of length $k$. As a result, $\mathrm{tr}[\psi_{\mathrm{Rad}}^k]$ can be calculated in the planar limit very simply by way of free probability theory.

More precisely, we may define moment-generating functions for EOW, JT, and Rad:

$$ \begin{aligned} M_{\mathrm{JT}}(z) &= \sum_{k=1}^{\infty} \frac{Z_k}{Z_1^k} z^k, \\ M_{\mathrm{EOW}}(z) &= \sum_{k=1}^{\infty} m_k z^k, \\ M_{\mathrm{Rad}} &= \sum_{k=1}^{\infty} e^{S_0(k-1)} \mathrm{tr}[\psi_{\mathrm{Rad}}^k] z^k. \end{aligned} $$

Given a moment-generating function $M(z)$, which is a formal power series, recall that its S-transform is given by

$$ S(z) = \frac{1+z}{z} M^{-1}(z). $$

We use this to define the S-transforms $S_{JT}$, $S_{EOW}$ and $S_{Rad}$. From Theorem 3.2, we see that the relation between the moments in Eq. (A.8) implies
that these are related as

$$S_{\text{Rad}}(z) = \frac{1}{1+z} S_{\text{JT}}(z) S_{\text{EOW}}(z). \qquad (A.9)$$

This means that, at the phase transition where $n \approx e^{S_0}$, the spectrum of $\psi_{\text{Rad}}$ can be described as a free product of the spectra of the end of the world brane, the JT gravity spectrum and a Marchenko-Pastur distribution.

### A.3.1. A Recursion Relation for the Resolvent.
Given a moment-generating function $M(z)$, we may also define the resolvent function $R(z)$ by

$$R(z) = \frac{1}{z} \left( 1 + M \left( \frac{1}{z} \right) \right).$$

To relate to previous results, we consider the case where the entanglement with the radiation is maximally entangled. In this case, $S_{\text{EOW}}(z) = 1$ and $S_{\text{Rad}}(z) = \frac{1}{1+z} S_{\text{JT}}(z)$. By definition of the S-transform and setting $z \to M_{\text{Rad}}(z)$, this implies

$$\frac{1}{1 + M_{\text{Rad}}(z)} S_{\text{JT}}(M_{\text{Rad}}(z)) = \frac{1 + M_{\text{Rad}}(z)}{M_{\text{Rad}}(z)} z,$$

which we may rewrite as (again using the definition of the S-transform):

$$M_{\text{Rad}}(z) = M_{\text{JT}}[z(1 + M_{\text{Rad}}(z))].$$

In terms of the resolvent, this becomes

$$\begin{aligned} R(z) &= \frac{1}{z} + \frac{1}{z} M_R \left( \frac{1}{z} \right) \\ &= \frac{1}{z} + \frac{1}{z} M_{\text{JT}}(R(z)) \\ &= \frac{1}{z} + \sum_{k=1}^{\infty} \frac{Z_k}{Z_1^k} \frac{R(z)^k}{z}, \end{aligned}$$

which is a recursion relation previously derived in [60] by a diagrammatic argument. More generally, we can interpret Eq. (A.9) as a (complicated) recursion relation that directly generalizes the above recursion relation.

## B. Random Tensor Network States and Split Transfer Protocols

Our results involving general background states are closely related to the quantum-information-theoretic task of split transfer introduced in [26], which can be understood as a variant of quantum state merging. The standard setup is as follows: two parties, Alice and Bob, share a state $\phi_{ABC_1...C_m}$, with Alice controlling A, Bob controlling B, and the systems $C_1, \dots, C_m$ being m “帮手.” Let R be a reference system and $\phi_{ABRC_1...C_m}$ a purification of $\phi$. Initially, the state is shared not only by Alice and Bob, but also with all the helper systems $C_i$. The goal of split transfer is to try to redistribute the state
to Alice, Bob, and R, using local quantum operations and classical communication (LOCC) between Alice, Bob and the helper systems, and possibly with the assistance of additional maximally entangled states.

A split transfer protocol consists of

1. A partitioning of the set of the helper systems: $T_A \sqcup T_B = \{C_1, \dots, C_m\}$.

2. For each $C_i \in T_A$, a number $K_{A,i}$ of shared maximally entangled qubits between Alice and $C_i$, and for each $C_i \in T_B$, a number $K_{B,i}$ of shared maximally entangled qubits between Bob and $C_i$.

3. An LOCC operation between Alice, Bob and the helper systems, such that after applying the protocol, Alice and Bob share a state $\psi_{ABRC_1\dots C_m}$, which is close to $\phi_{ABRC_1\dots C_m}$, and is such that Alice possesses systems A and $T_A$, while Bob controls B and $T_B$. Moreover, after applying the protocol they may be in possession of a number $L_{A,i}$ or $L_{B,i}$ of (approximately) maximally entangled qubits between $C_i$, and, respectively, A or B.

In this case, we say that the split transfer protocol has *entanglement costs* $K_{A,i} - L_{A,i}$ for all $C_i \in T_A$ and $K_{B,i} - L_{B,i}$ for all $C_i \in T_B$. A precise definition can be found as Definition 14 in [26].

Intuitively, the helper systems need to transfer their correlations with R to Alice and Bob, but without touching R. For instance, a simple protocol would be that the helpers simply teleport their full system to either Alice or Bob, consuming EPR pairs, leading to large entanglement costs. We can construct a potentially much more efficient protocol by way of random measurements, as detailed in Proposition 16 of [26]. Roughly speaking, such a protocol functions because random measurements have the effect of *decoupling* the helper systems from R. The helpers perform simultaneous random measurements on their systems and send the results of their measurements to Alice and Bob. Then, Alice and Bob can use their share of the global state and their portions of the maximally entangled states to perform a decoding operation conditioned on the results of the random measurements. The state they receive will be a purification of $\psi_{ABC_1\dots C_m}$, which will then be equivalent to the original $\psi_{ABRC_1\dots C_m}$ up to local isometries. The way we set up the split transfer protocol above was in a one-shot fashion: We get a single copy of $\phi$ and need to determine the optimal entanglement cost for the protocol.

One can also consider asymptotic variants of split transfer, where one has many copies available and aims to achieve an optimal transfer rate. An example application is the *entanglement of assistance*. Suppose that Alice, Bob, and the helper systems $C_i$ get many copies of a pure state $\phi$. At what rate can they distill maximally entangled pairs between Alice and Bob, if Alice and Bob are allowed to perform LOCC operations with all the helper systems? In this case, the answer is that the rate is given by

$$ \min_{T_A} S(AT_A)_{\phi}, $$
that is, by minimizing the entanglement entropy over all bipartitions. This rate is reminiscent of the importance of minimal cuts in a random tensor network, and the connection was explored in [40].

To see how the task of split transfer relates to random tensor networks, we consider three boundary regions, A and B, under the control of Alice and Bob, and the purifying system R. Each of the bulk vertices on the network correspond to a “helper” party. We would like to know whether there exists a protocol in which the assisting parties are allowed to perform local operations and classical communication (LOCC) such that the state $\phi_{VR}$ is redistributed into a state $\rho_{ABR}$ held by Alice and Bob that can be transformed by local isometries, acting only on A and B, to a state close to $\phi_{VR}$. The protocol given in [26] consists of simultaneous random measurements by each of the helpers. This precisely corresponds to the random projections performed in constructing the random tensor network state with this background state!

In this light, we can interpret Theorem 4.4 as a result on split transfer. Let us assume that $\phi \in \mathcal{P}_-(ABRC_1...C_n)$, denote the associated random tensor network state by $\rho_{ABR}$, and choose a partitioning $T_A \sqcup T_B$ of the assisting (bulk) parties. Since $H_2(A|B)_{\phi|\phi} \ge H_{\min}(A|B)_{\phi}$, Theorem 4.4 directly yields

**Theorem.** Suppose that

$$H_{\min}(S_A|BRT_B)_{\phi} \ge K_1 \qquad (B.1)$$

for all non-empty subsets $S_A \subseteq T_A$ and

$$H_{\min}(S_B|ART_A)_{\phi} \ge K_2 \qquad (B.2)$$

for all non-empty subsets $S_B \subseteq T_B$. Then,

$$\begin{aligned} \mathbb{E} \min_{V_A, V_B} \|(V_A \otimes V_B \otimes I_R) \rho(V_A^\dagger \otimes V_B^\dagger \otimes I_R) - \phi_{ABRC_1...C_n}\|_1 \\ = \mathcal{O}((2^{-\frac{1}{4}K_1} + 2^{-\frac{1}{4}K_2}). \end{aligned} \qquad (B.3)$$

where the minimum is over isometries $V_A : \mathcal{H}_A \to \mathcal{H}_{AT_A}$ and $V_B : \mathcal{H}_B \to \mathcal{H}_{BT_B}$.

This result shows that if $K_1$ and $K_2$ are sufficiently large, then after measurement in a random basis, the state possessed by Alice and Bob can, with high probability, be used to approximately reconstruct $\phi$ by acting with local isometries on the systems of Alice and Bob.

Suppose we fix values for $K_1$ and $K_2$. If the conditions in Eqs. (B.1) and (B.1) are not satisfied for the initial state $\phi$, we can use another state where we have added an appropriate number of maximally entangled Bell pairs between the assisting parties, increasing the entanglement cost of the protocol.<fnref n="5" /> An interesting open question in this context is whether one can generalize this result using smooth entropies in Eqs. (B.1) and (B.1). As alluded to in Sect. 4, the problem is that for a general state $\phi$, one would need to perform

<footnote n="5">⁵In fact, the protocol in [26] is slightly more general than what we describe; rather than measuring a random state one could also measure a random projection of rank greater than 1. This can be used to obtain EPR pairs between the helpers and Alice and Bob to get nonzero $L_{A,i}$ and $L_{B,i}$.</footnote>
simultaneous smoothing for all the relevant subsystems, which remains an open problem.

There is an alternative approach, in which it is straightforwardly possible to use smooth entropies [26]. In this approach, one merges each party in $T_A$ one by one, and similarly for $T_B$. That is, we choose some ordering $T_A = \{1, \dots, m\} = [m]$ and we apply a sequence of state merging protocols where we merge the state in $m$ steps, where a single step merges $A \cup T_A \setminus [i-1]$ into $A \cup T_A \setminus [i]$. In this case, it is not hard to see that, if we allow some error the entanglement cost is determined by the smooth conditional entropies $H_{\min}^{\epsilon}(\{i+1\}|BR[i]T_B)_{\phi}$. We perform a similar protocol for $B$ and the assisting $T_B$ systems.

## B.1. Split Transfer and Recovery in Holography

Split transfer is closely related to *subregion-subregion duality*, or *entanglement wedge reconstruction*, in holography. Consider an asymptotically AdS, stationary semiclassical geometry which is dual to a boundary CFT state $\rho$. We fix a time-reversal invariant spatial slice and partition the boundary into $A$ and $\bar{A}$. Let $\gamma_A$ be the minimal surface for $A$, and recall that $\partial\gamma_A = \partial A$. The region enclosed by $A$ and $\gamma_A$ is the *entanglement wedge* of $A$, which we denote by $\Gamma_A$. The claim of entanglement wedge reconstruction is that if we act with a low-energy local bulk operator in the entanglement wedge of $A$, we can reconstruct the action of this operator as a corresponding operator acting on the boundary system $A$.

One way to make this more precise is by considering a *code subspace* of bulk states $S$, which can be thought of as a set of states obtained by acting with low-energy operators on a fixed semiclassical spacetime. The action of the operators is small, in the sense that they do not create a significant backreaction that changes the geometry. We then introduce a reference system $R$ of the same dimension as $S$. In this framework, the AdS/CFT correspondence describes an encoding of the bulk into the boundary, taking a bulk state $\phi_{\Gamma_A \Gamma_{\bar{A}} R}$ to a boundary state $\rho_{A \bar{A} R}$. In this setup, the claim of entanglement wedge reconstruction is that we can act with an isometry on $A$ to recover $\phi_{\Gamma_A R}$. (And $\Gamma_A$ is actually the maximal such region.) This situation corresponds to quantum state merging—there is a single decoder $A$. However, in AdS/CFT, one usually requires the stronger condition of *complementary entanglement wedge reconstruction*, in which the entanglement wedge for $\bar{A}$ is also the complement of $\Gamma_A$, so that $\phi_{\Gamma_{\bar{A}} R}$ is recoverable from $\bar{A}$. This stronger requirement with two decoders is closely related to split transfer. These ideas and the precise relation to quantum information theory remain an active area of research, e.g., [6, 7, 36, 40].

In particular, [6, 7, 14] have argued that one-shot quantum information is the correct framework to understand entanglement wedge reconstruction, as holography is fundamentally a one-shot setting—we are provided with a single copy of a gravitational or CFT state, rather than asymptotically many copies.
The limit of large effective central charge, and hence small $G_N$, does reproduce certain aspects of the many-copy limit, as we reviewed in “Appendix A.1.” However, the distinction between the one-shot and asymptotic regimes becomes apparent in the presence of large bulk entropy, where the bulk entropy, and hence $R$, is large. Such situations arise when studying, for example, the black hole information paradox. In this case, where we assume we have a bulk state $\phi$, the minimal surface prescription in Eq. (A.3) is replaced by quantum extremal surface prescription:

$$H(\rho_A) = \min \mathrm{ext}_{\gamma_A} \left\{ \frac{|\gamma_A|}{4G_N} + H(\Gamma_A)_{\phi} \right\},$$

where we minimize over extremal surfaces $\gamma_A$, and we minimize the joint contribution of the area of $\gamma_A$ and the bulk entropy contained in the associated entanglement wedge $\Gamma_A$. This formula has a natural tensor network interpretation: Consider a random tensor network state with background state $\phi = \phi_{V^{(b)}R} \otimes \phi_{V^{(l)}}$, where $\phi_{V^{(b)}R}$ is a general background state (accounting for bulk entropy) and $\phi_{V^{(l)}}$ is a tensor product of link states on a graph $G = (V, E)$. Let us take maximally entangled link states with dimension $D$. Then, for some cut $\Gamma_A$ with edge set $\gamma_A$, we have

$$H(\phi_{\Gamma_A}) = \log(D) |\gamma_A| + H(\phi_{\Gamma_A}^{(b)}),$$

and we may hope that minimization over this quantity along the cuts gives a good approximation to the entropy. Whether such a prescription is valid depends on the structure of the background state $\phi$. A proposal put forth in [6] is that the surface $\gamma_A$ with entanglement wedge $\Gamma_A$ gives the max-entanglement wedge, if $\Gamma_A$ is the largest region such that, for any other surface $\delta_A$ homologous to $A$, with $\Delta_A$ the region enclosed by $A$ and $\delta_A$, and where $\Delta_A$ is contained in $\Gamma_A$, it holds that

$$H_{\min}^{\epsilon}(\Gamma_A \setminus \Delta_A | \Gamma_A^c R) \gg \frac{|\gamma_A| - |\delta_A|}{4G_N}.$$

In this case, $\Gamma_A$ should be the largest region which can be (approximately) reconstructed from $A$. Again, one can think of a random tensor network where the background state is a tensor product of a bulk state and a maximally entangled link state of dimension $D$ with $\log(D) = \Theta(G_N^{-1})$ along the discretization of the space. Then, this condition is (apart from the simultaneous smoothing problem) equivalent to Eq. (4.13). Enforcing complementary reconstruction, in which $\Gamma_A$ is the max-entanglement wedge for $A$, and its complement $\Gamma_A^c$ is the max-entanglement wedge for $\bar{A}$, allows us to interpret the holographic encoding of the bulk state into the boundary as a version of one-shot split transfer. See [6] for a detailed discussion of this proposal for holographic systems.

### C. Joint Smoothing of Link States

In this section, we provide a proof of Lemma 4.5. In our application of one-shot entropy estimates in Sects. 4.3 and 4.4, we needed to jointly smooth over different subsystems for the background state $\phi$. For general background states,
joint smoothing is an open problem [25,31]; however, if the background state is a product of link states as in Eq. (2.2), we can straightforwardly do so. In order to prove Lemma 4.5, we first need some intermediate lemmas. These lemmas involve quantum states which can be expanded in some preferential basis with positive coefficients. Therefore, for Lemmas C.1 to C.3 and C.5, we assume that each quantum system has a preferential basis, which we assume without loss of generality to be the standard basis.

**Lemma C.1.** Suppose that $\phi \in \mathcal{P}_{\le}(XST)$ can be written as

$$ \phi_{XST} = \sum_{i} |i\rangle\langle i| \otimes \phi_{ST,i} $$

where each $\phi_{ST,i}$ is a pure state such that $|\phi_{ST,i}\rangle$ can be written with positive coefficients in the standard basis. If

$$ \tilde{\phi}_{XST} = \sum_{i} |i\rangle\langle i| \otimes \tilde{\phi}_{ST,i} $$

is such that each $\tilde{\phi}_{ST,i}$ is a pure state satisfying $|\tilde{\phi}_{ST,i}\rangle \le |\phi_{ST,i}\rangle$ elementwise, then

$$ H_{\max}(XS|T)_{\tilde{\phi}} \le H_{\max}(XS|T)_{\phi}. $$

*Proof.* We start by arguing that for the max-entropy

$$ H_{\max}(XS|T)_{\tilde{\phi}} = \max_{\sigma_T \in \mathcal{P}_{\le}(T)} \log F(\tilde{\phi}_{XST}, I_{XS} \otimes \sigma_T)^2 \quad (C.1) $$

we may choose $\sigma_T$ in the optimization problem with nonnegative matrix elements in the standard basis. Indeed, suppose that $\sigma_T$ realizes the maximum in Eq. (C.1). Then, we can write a spectral decomposition $\sigma_T = \sum_j p_j |e_j\rangle\langle e_j|$. Now let $|\tilde{e}_j\rangle = \sum_k |\langle e_j|k\rangle|k\rangle$. Then, $\tilde{\sigma}_T = \sum_j p_j |\tilde{e}_j\rangle\langle \tilde{e}_j| \in \mathcal{P}_{\le}(T)$ has non-negative matrix elements in the standard basis. Moreover,

$$ \begin{aligned} F(\tilde{\phi}_{XST}, I_{XS} \otimes \sigma_T)^2 &= \sum_i F(\tilde{\phi}_{ST,i}, I_S \otimes \sigma_T)^2 \\ &\le \sum_{i,j} p_j |\langle \tilde{\phi}_{ST,i} | I_S \otimes |e_j\rangle \langle e_j| |\tilde{\phi}_{ST,i}\rangle \\ &\le \sum_{i,j} p_j \langle \tilde{\phi}_{ST,i} | I_S \otimes |\tilde{e}_j\rangle \langle \tilde{e}_j| |\tilde{\phi}_{ST,i}\rangle \\ &= F(\tilde{\phi}_{XST}, I_{XS} \otimes \tilde{\sigma}_T)^2. \end{aligned} $$

Let $\sigma_T$ be optimal for Eq. (C.1) with nonnegative matrix elements. It is clear that if $|\tilde{\phi}_{ST,i}\rangle \le |\phi_{ST,i}\rangle$, then

$$ \begin{aligned} F(\tilde{\phi}_{XST}, I_{XS} \otimes \sigma_T)^2 &= \sum_i \langle \tilde{\phi}_{ST,i} | I_S \otimes \sigma_T | \tilde{\phi}_{ST,i} \rangle \\ &\le \sum_i \langle \phi_{ST,i} | I_S \otimes \sigma_T | \phi_{ST,i} \rangle \\ &= F(\phi_{XST}, I_{XS} \otimes \sigma_T)^2 \end{aligned} $$
and hence

$$ H_{\max}(XS|T)_{\tilde{\phi}} = \log F(\tilde{\phi}_{XST}, I_{XS} \otimes \sigma_T)^2 \le \log F(\phi_{XST}, I_{XS} \otimes \sigma_T)^2 \\ \le H_{\max}(XS|T)_{\phi}. $$

□

**Lemma C.2.** Suppose that $\phi \in \mathcal{P}_{\le}(XYST)$ can be written as

$$ \phi_{XYST} = \sum_{i} |i\rangle\langle i| \otimes |j\rangle\langle j| \otimes \phi_{ST,ij} $$

where each $\phi_{ST,ij}$ is a pure state such that $|\phi_{ST,ij}\rangle$ can be written with positive coefficients in the standard basis. If

$$ \tilde{\phi}_{XYST} = \sum_{i} |i\rangle\langle i| \otimes |j\rangle\langle j| \otimes \tilde{\phi}_{ST,ij} $$

is such that each $\tilde{\phi}_{ST,ij}$ is a pure state satisfying $|\tilde{\phi}_{ST,ij}\rangle \le |\phi_{ST,ij}\rangle$ element-wise, then

$$ H_{\max}(XS|YT)_{\tilde{\phi}} \le H_{\max}(XS|YT)_{\phi}. $$

*Proof.* If we have a state $\rho \in \mathcal{P}_{\le}(XYST)$ which is of the form

$$ \sum_{j} |j\rangle\langle j| \otimes \rho_{j} $$

where each $\rho_j \in \mathcal{P}_{\le}(XST)$, then (Proposition 4.6 in [73])

$$ H_{\max}(XS|YT) = \log \left( \sum_{j} 2^{H_{\max}(XS|T)_{\rho_j}} \right). $$

Together with Lemma C.1, this implies the result. □

The background states we consider are tensor products of link states, and they can be expressed with positive coefficients in a product basis along the half-edges. The following lemma will help us to show that when we perform smoothing we can retain some of this structure.

**Lemma C.3.** Suppose that $\phi \in \mathcal{P}_{\le}(XST)$ can be written as

$$ \phi_{XST} = \sum_{i} |i\rangle\langle i| \otimes \phi_{ST,i} $$

where each $\phi_{ST,i}$ is pure and has a Schmidt decomposition

$$ |\phi_{ST,i}\rangle = \sum_{j} \sqrt{\lambda_{i,j}} |jj\rangle $$

in the standard basis. Then,

$$ H_{\min}^{\epsilon}(XS|T)_{\phi} = H_{\min}(XS|T)_{\tilde{\phi}} $$

for a state $\tilde{\phi} \in \mathcal{P}_{\le}(XST)$ which is such that $P(\phi, \tilde{\phi}) \le \epsilon$ and

$$ \tilde{\phi}_{XST} = \sum_{i} |i\rangle\langle i| \otimes \tilde{\phi}_{ST,i} $$
with

$$|\tilde{\phi}_{ST,i}\rangle = \sum_j \sqrt{\tilde{\lambda}_{i,j}} |jj\rangle.$$

for some $0 \le \tilde{\lambda}_{i,j} \le 1$.

*Proof.* Let $Y$ be a copy of $X$ and let $R$ be an additional reference system of sufficiently large dimension. Then, we let

$$|\phi_{XYSTR}\rangle = \sum_{i,j} \sqrt{\lambda_{i,j}} |ii\rangle \otimes |jj\rangle \otimes |0\rangle$$

be a purification of $\phi_{XST}$. Now, by duality of smooth entropies

$$H_{\min}^{\varepsilon}(XS|T)_{\phi} = -H_{\max}^{\varepsilon}(XS|Y)_{\phi}$$

By Lemma 6.13 of [74], we can find a state

$$\bar{\phi}_{SXY} = \sum_{j} |j\rangle\langle j| \otimes \bar{\phi}_{XY,j}$$

such that $P(\phi, \bar{\phi}) \le \varepsilon$ and $H_{\max}^{\varepsilon}(XS|Y)_{\phi} = H_{\max}(XS|Y)_{\bar{\phi}}$. Now, an arbitrary purification of $\bar{\phi}_{SXY}$ will be of the form

$$|\bar{\phi}_{STXYR}\rangle = \sum_{j,k} |jk\rangle \otimes |\bar{\phi}_{XYR,jk}\rangle.$$

We see that

$$|\langle\phi|\bar{\phi}\rangle| = \sum_{i,j,k} \delta_{j,k} \sqrt{\lambda_{i,j}} |\langle ii| \otimes \langle 0|\rangle| |\phi_{XYR,jk}\rangle|$$

so, optimizing over the choice of purification $|\bar{\phi}\rangle$, by Uhlmann's theorem we find a purification of the form

$$|\bar{\phi}_{STXYR}\rangle = \sum_{j} |jj\rangle \otimes |\bar{\phi}_{XYR,j}\rangle$$

such that $F(\phi_{XYS}, \bar{\phi}_{XYS}) = |\langle\phi|\bar{\phi}\rangle|$. We define the projector $\Pi_{XY} = \sum_j |jj\rangle\langle jj|$. Then for any $\sigma_Y \in \mathcal{P}_\le(Y)$ we note that $(I_S \otimes \Pi_{XY})I_{XS} \otimes \sigma_Y(I_S \otimes \Pi_{XY}) \le I_{XS} \otimes \sigma'_Y$ where $\sigma'_Y = \sum_j \langle j|\sigma_Y|j\rangle |j\rangle\langle j|$ and hence

$$F((I_S \otimes \Pi_{XY})\bar{\phi}(I_S \otimes \Pi_{XY}), I_{XS} \otimes \sigma_Y) = F(\bar{\phi}, (I_S \otimes \Pi_{XY})I_{XS} \otimes \sigma_Y(I_S \otimes \Pi_{XY})) \le F(\bar{\phi}, I_{XS} \otimes \sigma'_Y).$$

If we let $|\phi'_{XYSTR}\rangle = (\Pi_{XY} \otimes I_{STR})|\bar{\phi}_{XYSTR}\rangle$ this implies that

$$H_{\max}(XS|Y)_{\phi'} \le H_{\max}(XS|Y)_{\bar{\phi}}.$$

Moreover, it is easy to see that $|\langle\phi'|\phi\rangle| = |\langle\bar{\phi}|\phi\rangle|$. Again using duality, we find that

$$H_{\min}^{\varepsilon}(XS|T)_{\phi} \le H_{\min}(XS|TR)_{\phi'}.$$

Finally, let

$$|\tilde{\phi}\rangle = (I_{XYST} \otimes \langle 0|)|\phi'\rangle$$
then by data processing (Theorem 5.7 of [73], note that data processing for the conditional min-entropy is valid for trace non-increasing completely positive maps) it holds that

$$H_{\min}(XS|TR)_{\phi'} \le H_{\min}(XS|T)_{\tilde{\phi}}.$$

By construction $|\langle\tilde{\phi}|\phi\rangle| = |\langle\phi'|\phi\rangle|$ and hence $P(\tilde{\phi}, \phi) \le \varepsilon$. Finally, by construction $\tilde{\phi}$ is of the desired form

$$|\tilde{\phi}_{STXY}\rangle = \sum_{i,j} \sqrt{\tilde{\lambda}_{i,j}} |ii\rangle \otimes |jj\rangle.$$

The following lemmas will be used in our joint smoothing construction to bound the purified distance of the smoothed state.

**Lemma C.4.** Suppose $\phi, \psi \in \mathcal{P}_{\le}(\mathcal{H})$ are pure states, and suppose

$$|\phi\rangle = \sum_i \sqrt{\lambda_i} |i\rangle \qquad |\psi\rangle = \sum_i \sqrt{\mu_i} |i\rangle$$

for some $\lambda_i, \mu_i \ge 0$. Then,

$$T(\phi, \psi) \le P(\phi, \psi) \le \sqrt{2 \sum_i |\lambda_i - \mu_i|}.$$

*Proof.* Let

$$\rho = \Delta(\phi), \qquad \sigma = \Delta(\psi)$$

where $\Delta$ is the completely dephasing channel, so

$$\rho = \sum_i \lambda_i |i\rangle\langle i| \qquad \sigma = \sum_i \mu_i |i\rangle\langle i|.$$

Then, since $\rho$ and $\sigma$ are diagonal in the same basis

$$P(\phi, \psi) = P(\rho, \sigma),$$

we then estimate the trace distance by

$$T(\rho, \sigma) \le \|\rho - \sigma\|_1 = \sum_i |\lambda_i - \mu_i|$$

and we apply the Fuchs-van de Graaff inequalities, Eq. (4.1), to estimate

$$T(\phi, \psi) \le P(\phi, \psi) = P(\rho, \sigma) \le \sqrt{2T(\rho, \sigma)} \le \sqrt{2 \sum_i |\lambda_i - \mu_i|}.$$

**Lemma C.5.** Suppose $\phi, \phi_j \in \mathcal{P}_{\le}(\mathcal{H})$ are pure states for $j \in [n]$, and suppose

$$|\phi\rangle = \sum_i \sqrt{\lambda_i} |i\rangle \qquad |\phi_j\rangle = \sum_i \sqrt{\lambda_{j,i}} |i\rangle$$

for some $\lambda, \lambda_{i,j} \ge 0$ and suppose $P(\phi, \phi_j) \le \varepsilon$ for all $j$. Then, if we let

$$|\tilde{\phi}\rangle = \sum_i \min_j \sqrt{\lambda_{j,i}} |i\rangle$$
it holds that $P(\phi, \tilde{\phi}) \le 2\sqrt{n\varepsilon}$.

Proof. Let

$$ \rho = \sum_i \lambda_i |i\rangle\langle i| \qquad \rho_j = \sum_i \lambda_{j,i} |i\rangle\langle i| $$

Then, $P(\rho, \rho_j) = P(\phi, \phi_j)$ and using the Fuchs-van de Graaff inequalities

$$ \begin{aligned} \sum_i \left| \lambda_i - \min_j \lambda_{j,i} \right| &\le \sum_j \sum_i \left| \lambda_i - \lambda_{j,i} \right| = \sum_j \| \rho - \rho_j \|_1 \\ &\le 2 \sum_j T(\rho, \rho_j) \le 2 \sum_j P(\phi, \phi_j) \le 2n\varepsilon. \end{aligned} $$

The result now follows from Lemma C.4. □

<figure><img src="image_26.png" /></figure>

Finally, we prove our main joint smoothing result. We again consider the setting of a tensor product of link states on a graph $G = (V, E)$, where we consider some boundary subsystem $A \subseteq V_\partial$. Recall that for a cut $\Gamma_A \in C(A)$ we define

$$ C_1(\Gamma_A) = \{ \Delta_A \in C(A) : \Delta_A \subsetneq \Gamma_A \} $$

$$ C_2(\Gamma_A) = \{ \Delta_A \in C(A) : \Gamma_A \subsetneq \Delta_A \}. $$

**Lemma 4.5.** Let $\phi \in \mathcal{P}_-(V)$ be a link state, $A \subseteq V_\partial$ a boundary subsystem and $\Gamma_A \in C(A)$ a cut for $A$. Then, there exists a pure state $\phi^\epsilon \in \mathcal{P}_-(V)$ which is such that

$$ P(\phi, \phi^\epsilon) \le 2 \left( \sqrt{|C_1(\Gamma_A)|} + \sqrt{|C_2(\Gamma_A)|} \right) \sqrt{\epsilon} $$

and it holds that for any $\Delta_A \in C_1(\Gamma_A)$

$$ H_{\min}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi^\epsilon} \ge H_{\min}^\epsilon(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_\phi $$

and for any $\Delta_A \in C_2(\Gamma_A)$

$$ H_{\min}(\Delta_A \setminus \Gamma_A | \Gamma_A)_{\phi^\epsilon} \ge H_{\min}^\epsilon(\Delta_A \setminus \Gamma_A | \Gamma_A)_\phi. $$

*Proof.* We may assume without loss of generality that $\phi = \bigotimes_{e \in E} \phi_e$ is such that each $\phi_e$ has Schmidt decomposition in the standard basis,

$$ |\phi_e\rangle = \sum_{i=1}^{D_e} \sqrt{\lambda_{e,i}} |ii\rangle $$

This means we may write

$$ |\phi\rangle = \sum_I \sqrt{\lambda_I} |I\rangle $$

where $I$ runs over all possible basis elements along each edge $I = \{i_e\}_{e \in E}$ and

$$ \lambda_I = \prod_{e \in E} \lambda_{e,i_e} \qquad |I\rangle = \bigotimes_{e \in E} |i_e i_e\rangle. $$
We also let

$$E_1 = \{e = (xy) \in E, x, y \in \Gamma_A\}$$

$$E_2 = \{e = (xy) \in E, x, y \in \Gamma_A^c\}.$$

so $E = E_1 \sqcup E_2 \sqcup \gamma_A$.

Consider a cut $\Delta_A \in \mathcal{C}_1(\Gamma_A)$. Let

$$X = \{(e, x) : e = (xy), x \in \Gamma_A \setminus \Delta_A, y \in \Delta_A\}$$

$$S = \{(e, x) : e = (xy), x \in \Gamma_A \setminus \Delta_A, y \in \Delta_A^c\}$$

$$T = \{(e, x) : e = (xy), x \in \Gamma_A^c, y \in \Delta_A^c\}$$

Then, we see that $H_{\min}^{\varepsilon}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi} = H_{\min}^{\varepsilon}(XS|T)_{\phi}$ and we can write (as $\phi$ is a product state)

$$\phi_{XST} = \phi_X \otimes \phi_{ST}.$$

Here, $\phi_{ST}$ is pure, whereas $\phi_X$ is diagonal in the standard basis along the half-edges in $X$. From Lemma C.3 (applied with $\phi_{ST,i} = \phi_{ST}$ for each $i$), we find that we obtain a state $\phi^{\Delta_A, \varepsilon}$ which is such that $P(\phi^{\Delta_A, \varepsilon}, \phi) \le \varepsilon$ and

$$H_{\min}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi^{\Delta_A, \varepsilon}} \ge H_{\min}^{\varepsilon}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi}$$

and which can be written as

$$|\phi^{\Delta_A, \varepsilon}\rangle = \sum_I \sqrt{\lambda_I^{\Delta_A, \varepsilon}} |I\rangle.$$

for some coefficients $\lambda_I^{\Delta_A, \varepsilon}$. Moreover, we can write

$$|\phi^{\Delta_A, \varepsilon}\rangle = \sum_i |ii\rangle \otimes |\phi_{E_1, i}^{\Delta_A, \varepsilon}\rangle \otimes |\phi_{E_2}\rangle$$

where $i$ runs over all possible basis elements along the cut $\gamma_A$, $i = \{i_e\}_{e \in \gamma_A}$ and $|i\rangle = \bigotimes_{e \in \gamma_A} |i_e\rangle$, and the $\phi_{E_1, i}^{\Delta_A, \varepsilon}$ are pure states with positive coefficients in the standard basis. We now let

$$|\phi_1^\varepsilon\rangle = \sum_I \sqrt{\lambda_I^\varepsilon} |I\rangle$$

$$\lambda_I^\varepsilon = \min_{\Delta_A \in \mathcal{C}_1(\Gamma_A)} \lambda_I^{\Delta_A, \varepsilon}.$$

By construction, this state can be written as

$$|\phi_1^\varepsilon\rangle = \sum_i |ii\rangle \otimes |\phi_{E_1, i}^\varepsilon\rangle \otimes |\phi_{E_2}\rangle$$

By Lemma C.5, it holds that

$$P(\phi, \phi_1^\varepsilon) \le 2\sqrt{|\mathcal{C}_1(\Gamma_A)|} \varepsilon.$$
An analogous construction can be used to construct $\phi^{\Delta_A, \varepsilon}$ for $\Delta_A \in C_2(\Gamma_A)$, and taking the minimum over all such cuts, we get a state $\phi_2^\varepsilon$ of the form

$$|\phi_2^\varepsilon\rangle = \sum_i |ii\rangle \otimes |\phi_{E_1}\rangle \otimes |\phi_{E_2,i}^\varepsilon\rangle$$

which satisfies

$$P(\phi, \phi_2^\varepsilon) \le 2\sqrt{|C_2(\Gamma_A)|} \varepsilon.$$

We now define

$$|\phi^\varepsilon\rangle = \sum_i |ii\rangle \otimes |\phi_{E_1,i}^\varepsilon\rangle \otimes |\phi_{E_2,i}^\varepsilon\rangle.$$

We will now show that $\phi^\varepsilon$ has the desired properties. First of all, since $\phi$ is normalized, $F_*(\phi, \phi^\varepsilon) = F(\phi, \phi^\varepsilon)$ and $F(\phi, \phi^\varepsilon) = F(\phi_1^\varepsilon, \phi_2^\varepsilon)$ so

$$P(\phi, \phi^\varepsilon) = P(\phi_1^\varepsilon, \phi_2^\varepsilon) \le P(\phi, \phi_1^\varepsilon) + P(\phi, \phi_2^\varepsilon) \le 2 \left( \sqrt{|C_1(\Gamma_A)|} + \sqrt{|C_2(\Gamma_A)|} \right) \sqrt{\varepsilon}$$

using the fact that the purified distance is a metric.

Next, consider $\Delta_A \in C_1(A)$. We note that by duality $H_{\min}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi^\varepsilon} = -H_{\max}(\Gamma_A \setminus \Delta_A | \Delta_A)_{\phi^\varepsilon}$. We define the following subsystems of half-edges,

$$X' = \{(e, x) : e = (xy), x \in \Gamma_A \setminus \Delta_A, y \in \Gamma_A^c\}$$

$$Y' = \{(e, x) : e = (xy), x \in \Delta_A, y \in \Gamma_A^c\}$$

$$S' = \{(e, x) : e = (xy), x \in \Gamma_A \setminus \Delta_A, y \in \Gamma_A\}$$

$$T' = \{(e, x) : e = (xy), x \in \Delta_A, y \in \Gamma_A\}.$$

so $H_{\max}(\Gamma_A \setminus \Delta_A | \Delta_A)_{\phi^\varepsilon} = H_{\max}(X'S'|Y'T')_{\phi^\varepsilon}$. Next, we observe that by construction, $\phi^\varepsilon$ is of the form

$$\phi_{X'Y'S'T'}^{\varepsilon} = \sum_{i,j} |i\rangle\langle i| \otimes |j\rangle\langle j| \otimes \phi_{S'T',ij}^{\varepsilon}$$

where $i$ runs over all possible basis elements along $\gamma_A \setminus \delta_A$, $i = \{i_e\}_{e \in \gamma_A \setminus \delta_A}$ and $|i\rangle = \bigotimes_{e \in \gamma_A \setminus \delta_A} |i_e\rangle$, which forms a basis for $\mathcal{H}_X$; and $j$ runs over all possible basis elements along $\gamma_A \cap \delta_A$, $j = \{j_e\}_{e \in \gamma_A \cap \delta_A}$ and $|j\rangle = \bigotimes_{e \in \gamma_A \cap \delta_A} |j_e\rangle$, which forms a basis for $\mathcal{H}_Y$. Each $\phi_{S'T',ij}^\varepsilon$ is a pure state. Similarly,

$$\phi_{X'Y'S'T'}^{\Delta_A, \varepsilon} = \sum_{i,j} |i\rangle\langle i| \otimes |j\rangle\langle j| \otimes \phi_{S'T',ij}^{\Delta_A, \varepsilon}$$

where the $\phi_{S'T',ij}^{\Delta_A, \varepsilon}$ are pure states. Moreover, by construction $|\phi_{S'T',ij}^\varepsilon\rangle \le |\phi_{S'T',ij}^{\Delta_A, \varepsilon}\rangle$ elementwise in the standard basis. Therefore, by Lemma C.2 it holds that

$$H_{\max}(X'S'|Y'T')_{\phi^\varepsilon} \le H_{\max}(X'S'|Y'T')_{\phi^{\Delta_A, \varepsilon}}.$$

Finally, by duality $H_{\min}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi^{\Delta_A, \varepsilon}} = -H_{\max}(X'S'|Y'T')_{\phi^{\Delta_A, \varepsilon}}$ and we conclude

$$H_{\min}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi^\varepsilon} \ge H_{\min}(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi^{\Delta_A, \varepsilon}} \ge H_{\min}^\varepsilon(\Gamma_A \setminus \Delta_A | \Gamma_A^c)_{\phi}.$$
A similar argument is valid for $\Delta_A \in C_2(\Gamma_A)$, showing that

$$H_{\min}(\Delta_A \setminus \Gamma_A | \Gamma_A)_{\phi^{\epsilon}} \geq H_{\min}(\Delta_A \setminus \Gamma_A | \Gamma_A)_{\phi^{\Delta_A, \epsilon}} \geq H_{\min}^{\epsilon}(\Delta_A \setminus \Gamma_A | \Gamma_A)_{\phi}.$$

□

In the proof of Theorem 4.12, we need a similar smoothing lemma for a slightly different situation as stated in Lemma 4.11.

Recall the setup: We consider a graph $G = (V, E)$ and we have a set of boundary vertices $V_{\partial}$, and $A \subseteq V_{\partial}$. We denote by $\gamma_{A,1}$ the edges incident to $A$, and $\gamma_{A,2}$ the edges incident to $\bar{A}$, and we assume these sets do not intersect. We let $E_b = E \setminus (\gamma_{A,1} \cup \gamma_{A,2})$. For a cut $\Delta_A \in C(A)$, we let $Y^{\Delta_A}$ be the set of half-edges

$$Y^{\Delta_A} = \{(e, x) : e = (xy), x \in \Delta_A^c, y \in A\}.$$

**Lemma 4.11.** Let $\phi \in \mathcal{P}_{\le}(V)$ be a pure background state which is of the form $\phi = \phi_{\gamma_{A,1}} \otimes \phi_{\gamma_{A,2}} \otimes \phi_{E_b}$, where

$$\phi_{E_b} = \bigotimes_{e \in E_b} \phi_e$$

is a product state and the $\phi_{\gamma_{A,i}}$ have a Schmidt decomposition in the standard basis along the half-edges. Then, there exists a state $\phi^{\epsilon}$ which is such that

$$P(\phi, \phi^{\epsilon}) \leq 2\sqrt{2^{V_b}\epsilon}$$

and for all cuts $\Delta_A \in C(A)$ it holds that

$$H_{\min}(\Delta_A \setminus A | A)_{\phi^{\epsilon}} \geq H_{\min}^{\epsilon}(\Delta_A \setminus A | AY^{\Delta_A})_{\phi}.$$

*Proof.* The argument is much the same as the proof of Lemma 4.5. We again assume without loss of generality that each $\phi_e$ has Schmidt decomposition in the standard basis, and we may write

$$|\phi\rangle = \sum_I \sqrt{\lambda_I} |I\rangle$$

as in the proof of Lemma 4.5. Consider a cut $\Delta_A \in C(A)$. Then, we let

$$X = \{(e, x) : e = (xy), x \in \Delta_A, y \in \Delta_A^c\}$$

$$S = \{(e, x) : e = (xy), x \in \Delta_A, y \in \Delta_A \setminus A\}$$

$$T = \{(e, x) : e = (xy), x \in A, y \notin A\} \cup Y^{\Delta_A}$$

so $H_{\min}^{\epsilon}(\Delta_A \setminus A | AY^{\Delta_A})_{\phi} = H_{\min}^{\epsilon}(XS|T)_{\phi}$. Moreover, by the structure of the state, $\phi_{XST} = \phi_X \otimes \phi_{ST}$, where $\phi_{ST}$ is pure and $\phi_X$ is diagonal in the standard basis. So, from Lemma C.3 we obtain a state

$$|\phi^{\Delta_A, \epsilon}\rangle = \sum_I \sqrt{\lambda_I^{\Delta_A, \epsilon}} |I\rangle$$
for some real coefficients $\lambda_I^{\Delta_A, \varepsilon} \ge 0$ which is such that $P(\phi, \phi^{\Delta_A, \varepsilon}) \le \varepsilon$ and $H_{\min}(XS|T)_{\phi^{\varepsilon}} \ge H_{\min}^{\varepsilon}(XS|T)_{\phi}$. As before, we define

$$|\phi^{\varepsilon}\rangle = \sum_I \sqrt{\lambda_I^{\varepsilon}} |I\rangle \\ \lambda_I^{\varepsilon} = \min_{\Delta_A \in C(A)} \lambda_I^{\Delta_A, \varepsilon}.$$

By Lemma C.5 $P(\phi, \phi^{\varepsilon}) \le 2\sqrt{|C(A)|\varepsilon} = 2\sqrt{2|V_b|\varepsilon}$. If we define the following subsystems

$$X' = \{(e, x) : e = (xy), x \in \Delta_A, yA\}$$

$$Y' = \{(e, x) : e = (xy), x \in \Delta_A^c, yA\}$$

$$S' = \{(e, x) : e = (xy), x \in \Delta_A, y \notin A\}$$

$$T' = \{(e, x) : e = (xy), x \in \Delta_A^c, y \notin A\}$$

then $\phi_{X'S'T'}^{\varepsilon}$ can be written as

$$\phi_{X'Y'S'T'}^{\varepsilon} = \sum_{i,j} |i\rangle\langle i| \otimes |j\rangle\langle j| \otimes \phi_{S'T',ij}^{\varepsilon}$$

where $i$ runs over all possible basis elements along $\gamma_{A,1} \setminus \delta_A$, $i = \{i_e\}_{e \in \gamma_{A,1} \setminus \delta_A}$ and $|i\rangle = \bigotimes_{e \in \gamma_{A,1} \setminus \delta_A} |i_e\rangle$, which forms a basis for $\mathcal{H}_{X'}$; and $j$ runs over all possible basis elements along $\gamma_{A,1} \cap \delta_A$, $j = \{j_e\}_{e \in \gamma_{A,1} \cap \delta_A}$ and $|j\rangle = \bigotimes_{e \in \gamma_{A,1} \cap \delta_A} |j_e\rangle$, which forms a basis for $\mathcal{H}_{Y'}$. The $\phi_{S'T',ij}^{\varepsilon}$ are pure states. We can therefore apply Lemma C.1, and

$$H_{\max}(X'S'|T'Y')_{\phi^{\varepsilon}} \le H_{\max}(X'S'|T'Y')_{\phi^{\Delta_A, \varepsilon}}$$

and hence, by applying duality and data processing

$$\begin{align*} H_{\min}(\Delta_A \setminus A|A)_{\phi^{\varepsilon}} &= -H_{\max}(\Delta_A \setminus A|T'Y')_{\phi^{\varepsilon}} = -H_{\max}(X'S'|T'Y')_{\phi^{\varepsilon}} \\ &\ge H_{\max}(X'S'|T'Y')_{\phi^{\Delta_A, \varepsilon}} = H_{\min}(\Delta_A \setminus A|A)_{\phi^{\Delta_A, \varepsilon}} \\ &\ge H_{\min}(\Delta_A \setminus A|AY^{\Delta_A})_{\phi^{\Delta_A, \varepsilon}} \ge H_{\min}^{\varepsilon}(\Delta_A \setminus A|AY^{\Delta_A})_{\phi} \end{align*}$$

□

**Remark C.6.** We may make two observations on Lemmas 4.5 and 4.11. In both cases, if we construct the coefficients as $\min \lambda_I^{\varepsilon}$, $\lambda_I$ we find that we may assume the resulting state has coefficients in the basis $|I\rangle$ which are upper bounded by $\lambda_I$ (affecting only the constant factor in the upper bound for $P(\phi, \phi^{\varepsilon})$). Secondly, we note that

$$|\text{tr}[\phi] - \text{tr}[\phi^{\varepsilon}]| \le \sum_I |\lambda_I - \lambda_I^{\varepsilon}| = \mathcal{O}(\varepsilon) \quad (C.2)$$

rather than the naive $\mathcal{O}(\sqrt{\varepsilon})$ estimate.
## D. A Family of Metrics on the Symmetric Group

In analogy to the Cayley distance on the symmetric group $S_k$, we consider the following function $d_\rho$: $S_k \times S_k \to \mathbb{R}_{\ge 0}$ given some arbitrary density matrix $\rho \in \mathcal{P}_-(\mathcal{H})$:

$$d_{\rho}(\pi_1, \pi_2) = \sum_{l \in C(\pi_1^{-1} \pi_2)} (l-1)H_l(\rho) = - \sum_{l \in C(\pi_1^{-1} \pi_2)} \log \text{tr}[\rho^l].$$

As we saw in Sect. 2.1, this function is closely related to the replica trick for random tensor networks with non-trivial link states. In the case where $\rho$ is a maximally mixed state, this function is precisely proportional to the Cayley distance (2.15). In this appendix, we show that $d_\rho$ is a metric for any non-pure quantum state $\rho$. (If it is pure, then $d_\rho$ vanished identically.)

We say that two permutations $\alpha, \beta \in S_k$ are disjoint if any point not fixed by $\alpha$ is fixed by $\beta$ and vice versa.

**Lemma D.1.** The function $d_\rho$ defines a metric on $S_k$ for any state $\rho$ which is not pure. Moreover, if $\rho$ does not have flat spectrum the following holds for all $\pi_1, \pi_2, \pi_3 \in S_k$:

$$d_{\rho}(\pi_1, \pi_2) + d_{\rho}(\pi_2, \pi_3) = d_{\rho}(\pi_1, \pi_3)$$

if and only if $\pi_1^{-1}\pi_2$ and $\pi_2^{-1}\pi_3$ are disjoint permutations.

*Proof.* The fact that $d_\rho(\pi_1, \pi_2) = 0$ if and only if $\pi_1 = \pi_2$ follows from the assumption that $\rho$ is not pure. The symmetry $d_\rho(\pi_1, \pi_2) = d_\rho(\pi_2, \pi_1)$ is clear from $C(\pi_1^{-1}\pi_2) = C(\pi_2^{-1}\pi_1)$. Thus, the only non-trivial property we have to show in order for $d_\rho$ to be a metric is the triangle inequality. We let $d = \dim(\mathcal{H})$ and we write $\text{spec}(\rho) = \{\lambda_i\}_{i=1}^d$. Then, by letting $\alpha = \pi_1^{-1}\pi_2$ and $\beta = \pi_2^{-1}\pi_3$ the triangle inequality

$$d_{\rho}(\pi_1, \pi_2) + d_{\rho}(\pi_2, \pi_3) \ge d_{\rho}(\pi_1, \pi_3)$$

is equivalent to

$$\prod_{l \in C(\alpha)} \left( \sum_{i=1}^{d} \lambda_i^l \right) \prod_{m \in C(\beta)} \left( \sum_{i=1}^{d} \lambda_i^m \right) \le \prod_{n \in C(\alpha\beta)} \left( \sum_{i=1}^{d} \lambda_i^n \right). \quad (D.1)$$

Moreover, we need to show that if the spectrum is not flat, we have equality if and only if $\alpha$ and $\beta$ are disjoint permutations. It suffices to show this for the case where $\beta$ is a cycle, as we can write $\beta$ as a product of disjoint cycles in general and iteratively apply the result for the case where $\beta$ is a cycle. We write $\beta = (i_1 \dots i_m)$ and $\beta_q = (i_1 \dots i_q)$ for $q \le m$ and we let $\alpha_q = \alpha\beta_q$ and $\alpha_0 = \alpha$.

We can then find a unique sequence of numbers $q_0 = 0 < q_1 < q_2 < \dots < m$ such that, if one compares $\alpha_{q_j}$ and $\alpha_{q_{j+1}}$, then either $q_{j+1} - q_j + 1$ cycles have merged into a single cycle, or a single cycle has split into $q_{j+1} - q_j + 1$ cycles. Moreover, these two operations are alternating in the sense that if $\alpha_{q_j}$ to $\alpha_{q_{j+1}}$ is a merge, then $\alpha_{q_{j+1}}$ to $\alpha_{q_{j+2}}$ is a split and vice versa. Indeed, $\beta_{q+1} = \beta_q(i_q i_{q+1})$, so $\alpha_{q+1} = \alpha_q(i_q i_{q+1})$. If $i_q$ and $i_{q+1}$ are in different cycles
in $\alpha_q$, then applying $(i_q, i_{q+1})$ merges these two cycles, and if $i_q$ and $i_{q+1}$ are in the same cycle in $\alpha_q$, then applying $(i_q, i_{q+1})$ splits this cycle into two cycles. Let

$$J = \{j : \alpha_{q_j} \text{ to } \alpha_{q_{j+1}} \text{ is a merge}\}$$

and for $j \in J$ let $m_j = q_{j+1} - q_j + 1$, then $\sum_{j \in J} m_j \le m$. Now, it is clear that for any collection $n_1, \dots, n_r$ of numbers with $\sum_{j=1}^r n_j \le n$ it holds that

$$\prod_{j=1}^{r} \left( \sum_{i=1}^{d} \lambda_{i}^{n_{j}} \right) \geq \sum_{i=1}^{d} \lambda_{i}^{n} \qquad (D.2)$$

where the inequality is strict unless $r = 1$ and $n_1 = n$ (since we assume that $\rho$ is not pure). In particular (recall that we assumed $\beta$ to be a cycle of length $m$), we may estimate the left-hand side of Eq. (D.1) by

$$\prod_{l \in C(\alpha)} \left( \sum_{i=1}^{d} \lambda_{i}^{l} \right) \left( \sum_{i=1}^{d} \lambda_{i}^{m} \right) \leq \prod_{l \in C(\alpha)} \left( \sum_{i=1}^{d} \lambda_{i}^{l} \right) \prod_{j \in J} \left( \sum_{i=1}^{d} \lambda_{i}^{m_j} \right). \qquad (D.3)$$

with equality if and only if $\{m_j\}_{j \in J} = \{m\}$. (So $\beta$ only merges cycles.) We will next argue that for $j \notin J$

$$\prod_{l \in C(\alpha_{q_j})} \left( \sum_{i=1}^{d} \lambda_i^l \right) \leq \prod_{l \in C(\alpha_{q_{j+1}})} \left( \sum_{i=1}^{d} \lambda_i^l \right). \qquad (D.4)$$

and for $j \in J$

$$\prod_{l \in C(\alpha_{q_j})} \left( \sum_{i=1}^{d} \lambda_i^l \right) \left( \sum_{i=1}^{d} \lambda_i^{m_j} \right) \leq \prod_{l \in C(\alpha_{q_{j+1}})} \left( \sum_{i=1}^{d} \lambda_i^l \right) \qquad (D.5)$$

with equality if and only if the spectrum is flat or all the cycles that are merged are 1-cycles. Then, combining Eqs. (D.3), (D.4) and (D.5) we may conclude that Eq. (D.1) holds, with equality if and only if $\beta$ is a disjoint cycle from $\alpha$.

Equation (D.4) follows immediately from Eq. (D.2), so it remains to show Eq. (D.5). To this end we will apply Jensen's inequality. In Eq. (D.5) let us assume that $s$ cycles of lengths $l_1, \dots, l_s$ are merged into a cycle of length $l$, so $l = \sum_{p=1}^s l_p$. We let $l_{s+1} = s$. Let $f_p : [d] \to \mathbb{R}_{\ge 0}$ be the function defined by $i \mapsto \lambda_i^{l_p-1}$, for $p \in [s+1]$. Moreover, let

$$\eta_p = \frac{l_p - 1}{l_p - 1} \quad \text{for } p \in [s].$$

We consider expectation values over the probability measure on $[d]$ where $i$ has probability $\lambda_i$. Since $\eta_p \le 1$ for all $p$, by Jensen's inequality for $\eta_p \ne 0$

$$\left( \mathbb{E} f_p^{\frac{1}{\eta_p}} \right)^{\eta_p} \geq \mathbb{E} f_p \qquad (D.6)$$
with equality if and only if the spectrum is flat or $\eta_p = 1$. Note that for $\eta_p = 0$ (equivalently $l_p = 1$), $\mathbb{E}f_p = 1$. Using that

$$ \sum_{p=1}^{s+1} \eta_p = \sum_{p=1}^{s} \frac{l_p - 1}{l - 1} + \frac{s - 1}{l - 1} = \frac{l - s + s - 1}{l - 1} = 1 $$

and for $\eta_p \neq 0$

$$ \mathbb{E}f_p^{\frac{1}{\eta_p}} = \sum_{i=1}^{d} \lambda_i^l $$

we find that

$$ \prod_{p=1}^{s} \left( \sum_{i=1}^{d} \lambda_i^{l_p} \right) \sum_{i=1}^{d} \lambda_i^s = \prod_{p=1}^{s+1} \mathbb{E}f_p \le \prod_{p, \eta_p \ne 0} \left( \mathbb{E}f_p^{\frac{1}{\eta_p}} \right)^{\eta_p} = \sum_{i=1}^{d} \lambda_i^l $$

Here, we have equality if and only if the spectrum is flat or for all $p \in [s+1]$ it holds that $\eta_p = 1$ or $\eta_p = 0$. Since $\eta_p < 1$ for $p \in [s]$ and $l_{s+1} > 1$, this only happens if $l_p = 1$ for $p \in [s]$ and $s = l_{s+1} = l$ (in other words, if all the merged cycles are 1-cycles). Applying this with $s = m_j$ for each $j \in J$ proves Eq. (D.5). □

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Newton Cheng and Geoff Penington

Department of Physics, Center for Theoretical Physics

University of California

Berkeley

USA

Cécilia Lancien

Institut Fourier and CNRS

Université Grenoble Alpes

Gières

France

Geoff Penington

Institute for Advanced Study

Princeton

USA

Michael Walter

Faculty of Computer Science

Ruhr University Bochum

Bochum

Germany

Freek Witteveen

Department of Mathematical Sciences and QMATH

University of Copenhagen

Copenhagen

Denmark

e-mail: fw@math.ku.dk

Communicated by David Pérez-García.

Received: November 17, 2022.

Accepted: August 14, 2023.