By taking a bipartition $A|B$ with $A$ being one site $i$ and $B$ its complement, then we find $|\psi ⟩=|{\psi }_i⟩\otimes |{\tilde{\psi }}_B⟩$. By repeating the same procedure for all other sites $i$, we conclude $|\psi ⟩={\otimes }_i|{\psi }_i⟩$.

For odd system size $L\ge 3$, we may redefine $U_{\mathrm{CZX}}\to i{U}_{\mathrm{CZX}}$ so it squares to $+1$. In that case, the results for even $N$ follow accordingly, up to global phases.

For example, it is easy to check that some tensor products of GHZ-like states in the disconnected configuration of Fig. 4 are symmetric under the CZX symmetry.

We thank Chaoming Jian for pointing out the necessity of having such a proof.

We note that Ref. [72] conjectured that states like ${\rho }_{\infty ,+}$ are long-ranged entangled when the strong symmetry is anomalous.

We note that Ref. [75] discussed a type of “intrinsic mixed-state quantum topological order (TO).” The authors proved that such states cannot be two-way connected to any fully separable states and argued that because “it is widely believed that nonmodular anyon theories cannot be realized by local gapped Hamiltonians in 2D bosonic system,” then “this implies the lack of a pure-state counterpart of the mixed-state TO; thus it is indeed intrinsically mixed.” However, currently, it is unknown whether such states can be two-way connected to some gapless pure states, especially when the channels are generalized and are able to create or eliminate long-range classical correlations.

A more precise statement of the mixed anomaly would come from calculating the 3-cocycle $\omega$ from Eq. (36). Apart from permutation of labels, it is easy to check that $\omega (a,b,c)=\exp (i\pi a_1{b}_2{c}_3)$, for $a,b,c\in {\mathbb{Z}}_2^3$. This 3-cocycle is not a coboundary, and it matches with the expected low-energy term in the gauged SPT bulk’s Lagrangian, $\frac{1}{2}{A}_1{A}_2{A}_3\in H^3[{\mathbb{Z}}_2^3,U(1)]$ [78]. Importantly, this anomaly is mixed because restricting $\omega$ to any ${\mathbb{Z}}_2^2\simeq ({\mathbb{Z}}_2,{\mathbb{Z}}_2,0)\subset {\mathbb{Z}}_2^3$ subgroup gives the trivial 3-cocycle $\omega {|}_{{\mathbb{Z}}_2^2}=1$.

We note that after the first version of our manuscript appeared online, several works [81, 82] also independently studied the consequences of mixed strong-weak anomaly. The constraints obtained here are different from and complementary to those in Refs. [81, 82].

See, however, Refs. [89, 90] for recent discussions on “anomaly” defined in different ways, which can be nontrivial even for purely weak symmetries.

Recall that the mutual information bounds the distance between a correlated state ${\rho }_{AB}$ and its marginals ${\rho }_A$ and ${\rho }_B$, as $I_{\rho }(A:B)=S({\rho }_{AB}\parallel {\rho }_A\otimes {\rho }_B)\ge (1/2\ln (2)){\parallel {\rho }_{AB}-{\rho }_A\otimes {\rho }_B\parallel }_1^2$, where $S(·\parallel ·)$ is the relative entropy and ${\parallel ·\parallel }_1$ the trace norm [66, 96].

In Definition 8, we require $A$ to be a disk region, which is not the case for annulus $\partial R_{\tilde{w}}$ in dimensions $d\ge 2$. However, since it does not wrap around the base manifold nontrivially (as it is the boundary of a disk region $R$), we find it reasonable to assume exponentially decaying CMI for $A=\partial R_{\tilde{w}}$ as well. In particular, topologically ordered states do satisfy this extra assumption.

The channel can be constructed as follows. Starting from consider a tripartition $A$, $B$, $C$, we can trace out the boundary regions as in the proof of Theorem 5, which leaves behind a tripartite factorized state ${\rho }_{A^{-}}\otimes {\rho }_{B^{-}}\otimes {\rho }_{C^{-}}$ due to the vanishing MI. Because of the vanishing CMI, we can then recover the original $\rho$ using Petz maps at the boundary region—this precisely gives a local channel that prepares $\rho$ from a tripartite factorized state.

Strictly speaking, the proof in Ref. [101] was done in the $L\to \infty$ limit, assuming a “smooth” thermodynamic limit. However, it is not difficult to modify the proof for finite $L$. The more precise conclusion is that the weak translation symmetry would be spontaneously broken if both the following conditions are satisfied: (i) for $|x-y|\ll L$, $⟨Z_x{Z}_y⟩$ decays faster than $1/|x-y{|}^2$;

(ii) for $|x-y|\sim O(L)$, $L^2⟨Z_x{Z}_y⟩\to 0$. For example, the state $\rho \propto P_{Q=0}$ with only strong $SO(2)$ symmetry has $⟨Z_x{Z}_y⟩\sim 1/L$, violating the second condition, therefore is compatible with translation symmetry not being spontaneously broken.