Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking

The well-known pure cluster state on a closed chain of $2N$ qubits [42] can be expressed as

where

are decorated domain wall states [48]. Here, we define $\ket{Z=\pm 1}$ ($\ket{X=\pm}$) to be the eigenstates of $Z$ ($X$) with eigenvalues $\pm 1$. Note that we denote the eigenstates of $X$ with eigenvalues $\pm 1$ by $\ket{\pm}$ in this paper. The decorated domain wall states $\ket{\Psi_{\{z_{i}\}}}$ are constructed by first specifying a configuration $\{z_{i}\}$ which determines the $Z$ charge on the odd sites, and then “decorating” each domain wall with a state charged under $X$. More specifically, when $z_{i}\neq z_{i+1}$, the interceding even site $2i$ is fixed to $\ket{-}$, and to $\ket{+}$ otherwise.

The decohered cluster state is defined by taking the equal-weight incoherent mixture of the decorated domain wall states shown in Eq.˜3, rather than their coherent superposition. Defined on a closed chain of $2N$ qubits, the decohered cluster state reads [15]

As we will see later, it is exhibits nontrivial mixed-state SPT order.

We hereby discuss the notions of weak and strong symmetry applicable to mixed states [35, 36, 12, 13]. In particular, a weakly $G$-symmetric matrix $M$ satisfies $U_{g}MU_{g}^{\dagger}=w(g)M$ for all $g\in G$, where $G$ is an Abelian symmetry group and $w(g)\in U(1)$. On the other hand, a strongly $G$-symmetric matrix $M$ satisfies $U_{g}M=s_{\text{ket}}(g)M$ and $MU_{g}^{\dagger}=s_{\text{bra}}(g)M$ for all $g\in G$, where $s_{\text{ket}}(g)$, $s_{\text{bra}}(g)\in U(1)$. When $M$ is Hermitian, $s_{\text{ket}}(g)=s_{\text{bra}}(g)^{*}$.

Consider the following operators in the Hilbert space of $2N$ qubits on a closed ring,

The decohered cluster state $\rho_{\mathcal{C}}$ is a mixed-state SPT order under the direct product of a strong $\mathbb{Z}_{2}$ symmetry and a weak $\mathbb{Z}_{2}$ symmetry, as classified in Refs. [15, 16], where the strong $\mathbb{Z}_{2}$ symmetry (denoted as $\mathbb{Z}_{2}^{S}$ from now on) is generated by $S$, while the weak $\mathbb{Z}_{2}$ symmetry (denoted as $\mathbb{Z}_{2}^{W}$ from now on) is generated by $W$. We will use $s_{\text{ket}}$, $s_{\text{bra}}$, and $w$ (with argument $g$ suppressed) when we are referring to the charges under the strong symmetry $S$ and weak symmetry $W$.

For the decohered cluster state $\rho_{\mathcal{C}}$ defined in Eq. (3), we find that $W\rho_{\mathcal{C}}W^{\dagger}=\rho_{\mathcal{C}}$ and $S\rho_{\mathcal{C}}=\rho_{\mathcal{C}}S^{\dagger}=\rho_{\mathcal{C}}$, so that $\rho_{\mathcal{C}}$ lies in the $(s_{\text{ket}},s_{\text{bra}},w)=(+1,+1,+1)$ symmetry sector. There is an intuitive way to understand why $W$ acts as a weak symmetry while $S$ acts as a strong symmetry. Note that $S$ counts the parity of the number of domain walls in each decorated domain wall state. With periodic boundary conditions, the system’s topology enforces an even number of domain walls in every state, $S\ket{\Psi_{\{z_{i}\}}}=\ket{\Psi_{\{z_{i}\}}}$. This will be the case for both the bra states and the ket states in $\rho_{\mathcal{C}}$. We therefore see that $\rho_{\mathcal{C}}$ must be strongly symmetric with $s_{\text{ket}}=s_{\text{bra}}=+1$. On the other hand, $W$ acts on the decorated domain wall states as $W\ket{\Psi_{\{z_{i}\}}}=\ket{\Psi_{\{\overline{z_{i}}\}}}$, where $\overline{z_{i}}\vcentcolon=-z_{i}$. This transformation is only a symmetry if it acts simultaneously from both the bra and the ket sides, indicating that $W$ can only be a weak symmetry.

One physically motivated approach to define mixed-state phases is to define an equivalence class relation among the states, where the equivalence is defined by a short-time physical processes. This mimics the definition of ground states being in the same phase if they can be adiabatically perturbed to each other rapidly. This definition was put forward by Ref. [8], and we review their definition more rigorously in Appendix˜A. Informally, it states that for two mixed states $\rho$ and $\sigma$ to be in the same phase, there has to exist two time-independent, symmetric geometrically local Lindbladians $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ such that $\|e^{\mathcal{L}_{1}t}(\rho)-\sigma\|_{1}$ and $\|e^{\mathcal{L}_{2}t}(\sigma)-\rho\|_{1}$ are small, where $t$ is some short time (“fast-driven”) and where $\|X\|_{1}\vcentcolon=\text{Tr}[\sqrt{X^{\dagger}X}]$ is the trace distance. Note that the smallness of the trace distance implies the closeness of all the observables $\hat{O}$ between $\rho$ and $\sigma$, i.e., $|\text{Tr}[\rho O]-\text{Tr}[\sigma O]|\leq\|\rho-\sigma\|_{1}\cdot\|O\|_{\text{op}}$ by Hölder’s inequality, where $\|\cdot\|_{\text{op}}$ is the operator norm. Furthermore, this “fast-driven” condition implies stability of local observables to local perturbations in the Lindbladian terms [49, 8, 50].

One can also define the equivalence-class relation through quantum channels, analogous to the Lindbladian equivalence-class relation. That is, if there exist two symmetric finite-depth local quantum channels [15, 16] $\mathcal{E}_{1}$ and ${\mathcal{E}}_{2}$ such that $\|\mathcal{E}_{1}(\rho)-\sigma\|_{1}$ and $\|\mathcal{E}_{2}(\sigma)-\rho\|_{1}$ are small, then we say $\rho$ and $\sigma$ are in the same phase. Note that in both the Lindbladian and quantum channel equivalence-class relations, the bi-directional “fast-driven” condition is important, as the Lindbladian evolution and the quantum channel are usually not reversible.

Building on the quantum-channel equivalence-class relation, Refs. [15, 16] show that if the strong-string order parameter (an order parameter used to diagnose SPT orders, see Section˜II.5) in the state $\sigma$ is zero, then there does not exist a symmetric finite-depth channel $\mathcal{E}$ such that the string order parameter of $\mathcal{E}(\sigma)$ is of order one. That is to say, if $\rho$ has a string order parameter of order one, then $\|\mathcal{E}(\sigma)-\rho\|_{1}$ can never be small. We know from Refs. [15, 16] that $\rho_{\mathcal{C}}$ has a nonzero string order parameter (see also Section˜II.5 for more explanation on this). Thus, $\rho_{\mathcal{C}}$ is in a nontrivial mixed-state SPT order protected by $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$, as stated in Refs. [15, 16].

Here, we complement the above result by showing that $\rho_{\mathcal{C}}$ has nontrivial SPT order protected by $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ symmetry under the Lindbladian equivalence-class relation as well. Using Lieb-Robinson bounds [51, 52], we show that $\|e^{\mathcal{L}t}(\sigma)-\rho\|_{1}$ can never be small for a symmetric Lindbladian $\mathcal{L}$ within some short time $t$. The detailed statement and its proof are presented in Appendix˜A.

The above results can be interpreted as follows—the nontrivial SPT order of $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ is hard to build, which forbids a “fast-driven” symmetric Lindbladian or a symmetric finite-depth quantum channel from bringing a trivial state $\sigma$ close to $\rho_{\mathcal{C}}$, which is a nontrivial SPT state. On the other hand, it is interesting to know if $\rho_{\mathcal{C}}$ can be brought close to a trivial state in a short time or in a short depth. In Appendix˜B, we show that the string order parameter corresponding to the $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ symmetry defined in Eq. (4) is easy to destroy. In particular, we show that a simple trace channel and the corresponding trace Lindbladian can destroy the string order in depth one and in a short time, respectively. Interestingly, the destruction of the string order is accompanied by the “strong-to-weak” spontaneous symmetry breaking, which has received a lot of attention recently [27, 38, 39].

It is instructive to express $\rho_{\mathcal{C}}$ as a matrix product density operator, as it helps us expose the edge modes on an open chain, extract the projective representation, and calculate the string order parameters. We first define two families of matrices in the virtual space spanned by $\ket{v_{0}}=(1,0)^{T}$ and $\ket{v_{1}}=(0,1)^{T}$. These matrices are two-index tensors given by $A^{z,z^{\prime}}=\Pi_{z}\delta_{z,z^{\prime}}$ and $B^{x,x^{\prime}}=X^{\frac{1-x}{2}}\delta_{x,x^{\prime}}$, where $\Pi_{z}\vcentcolon=\delta_{z=+1}|v_{0}\rangle\langle v_{0}|+\delta_{z=-1}|v_{1}\rangle\langle v_{1}|$ . We can then define two four-index tensors:

where $z,z^{\prime},x,x^{\prime}\in\{+1,-1\}$ are the physical indices, $\alpha,\beta\in\{0,1\}$ are the virtual indices, and as above $Z\ket{Z=z}=z\ket{Z=z}$ and $X\ket{X=x}=x\ket{X=x}$. Abbreviating the configuration $\Lambda=(z_{1},x_{1},\cdots,z_{N},x_{N})$, we can then write $\rho_{\mathcal{C}}=\sum_{\Lambda,\Lambda^{\prime}}\rho_{\mathcal{C}}(\Lambda,\Lambda^{\prime})\outerproduct{\Lambda}{\Lambda^{\prime}}$, where

The sums over $\Lambda$ and $\Lambda^{\prime}$ in the definition of $\rho_{\mathcal{C}}$ each go over all $2^{2N}$ possibilities, while the restriction to decorated domain wall states is taken care of by the coefficients $\rho_{\mathcal{C}}(\Lambda,\Lambda^{\prime})$. Pictorially, we write

where we have used periodic boundary conditions. When tensors appear without indices explicitly denoted, we assume they are summed over with the corresponding bras and kets.

It is worth noting that the tensors defined in Eqs.˜5 and 6 have the following “pulling-through” relations:

Using these relations, it will be easy to show that the symmetry $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ forms a projective representation on the virtual space.

Let us briefly review the relationship between SPT order in 1d and projective representations. Pure-state SPT phases in 1d protected by a global symmetry $G$ are characterized and classified by edge modes transforming in a projective representation of $G$. The classification can be formulated in terms of the second cohomology group $H^{2}(G,U(1))$, which tabulates the inequivalent projective representations of $G$ with $U(1)$ phases. In Ref. [15], it was shown that mixed-state SPT phases protected by a combination of a strong $K$ symmetry and weak $G$ symmetry are classified by $\bigoplus_{p=1}^{2}H^{2-p}(G,H^{p}(K,U(1))$. In the present case of the decohered cluster state, we have $G=K=\mathbb{Z}_{2}$, so that the above classification evaluates to $H^{1}(\mathbb{Z}_{2},H^{1}(\mathbb{Z}_{2},U(1)))=\mathbb{Z}_{2}$. This means that there is a single nontrivial mixed-state SPT phase with symmetry $\mathbb{Z}_{2}^{\text{S}}\times\mathbb{Z}_{2}^{\text{W}}$, with respect to the definition given in Ref. [15]. And indeed a nontrivial SPT order can be constructed from the domain-wall construction between the strong $\mathbb{Z}_{2}$ and the weak $\mathbb{Z}_{2}$ symmetries such as $\rho_{\mathcal{C}}$.

To see that the decohered cluster state indeed realizes the $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ symmetry projectively at its boundary, we first construct the following family of density matrices on open boundary conditions:

where $\leavevmode\hbox to16.09pt{\vbox to14.7pt{\pgfpicture\makeatletter\hbox{\quad\lower-1.7649pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.95863pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {{}} {{{}{{}{}}{}}}{{ {}{}{}}}{{{{}}{{}}}}{{}}{{{ }}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {{}\pgfsys@rect{-1.5649pt}{-1.5649pt}{3.12979pt}{3.12979pt}\pgfsys@fillstroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.59938pt}{5.2979pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$\alpha$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}=\bra{v_{\alpha}}$, $\leavevmode\hbox to15.76pt{\vbox to14.7pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.7649pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{}{}\pgfsys@moveto{18.49411pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {{}} {{{}{{}{}}{}}}{{ {}{}{}}}{{{{}}{{}}}}{{}}{{{ }}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {{}\pgfsys@rect{26.88786pt}{-1.5649pt}{3.12979pt}{3.12979pt}\pgfsys@fillstroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.45276pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.18556pt}{5.2979pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$\beta$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}=\ket{v_{\beta}}$ and $\alpha,\beta\in\{0,1\}$. The states in Eq.˜10 span a 4-dimensional linear space with complex coefficients, where the physical density matrices correspond to the subspace of Hermitian matrices with unit trace. This space can encode two classical bits of information.

We then act with the symmetry operators on $\rho_{\alpha\beta}$. Akin to the pure state case, we see via the pulling-through relations Eq.˜9 that the global symmetry factors to a product of operators acting on each virtual edge state:

We have found that the symmetries reduce to an effective action on the edge Hilbert space, namely

As in the pure state case, these operators commute globally, but anti-commute if either edge is taken in isolation:

The matrices $X$ and $Z$ obtained by pulling the symmetry to each edge generate the algebra $\{\mathbbm{1},X,Z,XZ\}$, which forms a projective representation of the group $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. This is precisely what we mean by the statement that the $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ symmetry is realized projectively at the edge.

While conventional SPT orders do not spontaneously break symmetry and therefore do not admit local order parameters, they can be characterized by non-local string order parameters [53, 54, 55, 56, 57]. Importantly, the diagnostic of an SPT order is done by the “patterns of zeros” of various string order parameters [55]. We here briefly review the definition of a string order parameter in the pure state case. Suppose we have an Abelian on-site symmetry $G$ with symmetry operators

In general, a string order parameter that detects a nontrivial projective representation of the $G$ symmetry consists of a finite string of $U_{g}^{(i)}$ operators in a region $\mathcal{R}$ with operators $\mathcal{O}^{(L/R)}_{\alpha}$ appended to the left and right ends of the string which transform under conjugate irreps $\alpha$ and $\alpha^{*}$ of the other symmetry,

The string operators $\mathcal{S}_{g}$ let us define the string order parameter for a pure state $\ket{\psi}$, which is simply $\langle\psi|\mathcal{S}_{g}|\psi\rangle$.

One natural generalization of the string order parameter from pure states to mixed states is to measure the expectation value of the strong-string operator $\mathcal{S}_{nm}^{S}$ (defined below) in the state $\rho$:

where

for odd $n$ and $m$. The Roman numeral I appearing in the subscript indicates that the correlator is linear in the density matrix (i.e. Rényi-1), and distinguishes it from other correlators which we will define shortly which are quadratic in the density matrix. This definition has been used in prior works, and it has been shown that it is an order parameter for the $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ mixed-state SPT order [16]. If we try to define an analogous order parameter for the weak-string operator $\mathcal{S}^{W}_{n,m}\vcentcolon=\mathcal{S}_{n+1,m+1}^{S}$, then we find that it is trivially one because the weak-string operator acts from both the ket and the bra sides and $\left(\mathcal{S}^{W}\right)^{2}=\mathbbm{1}$, more specifically $\text{Tr}\left[\mathcal{S}_{nm}^{W}\rho\mathcal{S}_{nm}^{W}\right]=1$.

One way of generating a useful string correlator corresponding to the weak symmetry is to map density matrices to vectors in a doubled Hilbert space with twice the number of qubits and consider the string correlators in this doubled space [27]. The mapping of density matrices to vectors in the doubled Hilbert space is as follows:

where the subscripts $L$ and $R$ denote the left and the right Hilbert spaces and $|i\rangle$ and $|j\rangle$ denote some computational basis.

In the doubled Hilbert space, the symmetry operators become $S\otimes\mathbbm{1}$, $\mathbbm{1}\otimes S$ for the strong ket and the strong bra symmetries, and $W\otimes W$ for the weak symmetry. Here the first factor of the tensor product acts on the left (ket) Hilbert space and the second factor acts on the right (bra) Hilbert space. We denote the expectation values of an operator $\mathcal{M}$ in the doubled Hilbert space by

where $\mathcal{M}\left[\cdot\right]$ is a superoperator that corresponds to the operator $\mathcal{M}$ in the doubled Hilbert space. For example, $\langle\rho|A\otimes B|\rho\rangle=\text{Tr}\left[\rho^{\dagger}A\rho B^{T}\right]$.

Now, by substituting the various symmetry operators in the definition of the string operator, Eq.˜16, we find the various string correlators that we need. If $U_{g}^{(i)}=(S\otimes\mathbbm{1})^{(i)}$ on the even sites, and the end point operators are $\mathcal{O}_{\alpha}^{(L/R)}=Z$ on the odd sites $m$ and $n$, then we get the following correlator which we call the Rényi-2 strong string correlator:

where $\mathcal{S}_{nm}^{S}$ is defined as in Eq.˜18 and the Roman numeral II reflects the fact that the correlator is quadratic in the density matrix, or Rényi-2. Note that $(W\otimes W)^{\dagger}(Z_{n/m}\otimes\mathbbm{1})\left(W\otimes W\right)=-Z_{n/m}\otimes\mathbbm{1}$. Thus this choice of the end-point operators can in principle be used to detect nontrivial SPT order.

If we take the symmetry to be the weak symmetry in Eq.˜16, that is, $U_{g}^{(i)}=(W\otimes W)^{(i)}$ on the odd sites, and the end-point operators $\mathcal{O}_{\alpha}^{(L/R)}=Z$ on the even sites $n$ and $m$, then we get the Rényi-2 weak string correlator:

where $\mathcal{S}^{W}_{n,m}=\mathcal{S}^{S}_{n+1,m+1}$ as defined earlier. Note that $(S\otimes\mathbbm{1})^{\dagger}(Z_{n/m}\otimes Z_{n/m})(S\otimes\mathbbm{1})=-Z_{n/m}\otimes Z_{n/m}$.

Next, we construct two more correlators by taking the end point operators to be $\mathbbm{1}$. These are the Rényi-2 trivial strong-string correlator and the Rényi-2 trivial weak-string correlator:

where $\tilde{\mathcal{S}}^{S}_{nm}$ has the strong symmetry operators in the bulk and $\mathbbm{1}$ as the end point operators with $n,m$ restricted to be odd, while $\tilde{\mathcal{S}}^{W}_{nm}$ has the weak symmetry operators in the bulk and $\mathbbm{1}$ as the end point operators with $n,m$ restricted to be even.

While for a generic density matrix, the Rényi-1 and Rényi-2 type of correlators are independent, Ref. [27] shows that they are related to each other if the state is short-range entangled. The patterns of zeros used to diagnose $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ trivial and nontrivial SPT orders are summarized in Table 3.

Now we show, using tensor network diagrams, that $C_{\text{I},nm}^{S}=C_{\text{II},nm}^{S}=C_{\text{II},(n-1),(m-1)}^{W}=1$ for $n,m\in\text{odd}$ in the state $\rho_{\mathcal{C}}$. Acting on $\rho_{\mathcal{C}}$ with each of the string operators, we find

Because $\rho_{\mathcal{C}}$ is invariant under the action of the strings, the string order parameters will be identically one whether we chose to contract the physical legs of $\rho_{\mathcal{C}}$ according to the Rényi-1 or Rényi-2 conventions.

Using the pulling-through relations Eq.˜9 and the definition of $\rho_{\mathcal{C}}$ [see Eq.˜3], we can show that $\tilde{C}_{\text{I},nm}^{S}=\tilde{C}_{\text{II},nm}^{S}=\tilde{C}_{\text{II},nm}^{W}=0$ in the decohered cluster state.

We define the parent Lindbladian on an open chain of $2N$ qubits labelled by $1,2,\ldots,2N$ as follows:

where $\mathcal{D}[A](\rho)\vcentcolon=A\rho A^{{\dagger}}-\frac{1}{2}\{A^{{\dagger}}A,\rho\}$ denotes the standard Lindblad dissipator for a jump operator $A$ [58, 59]. For periodic boundary conditions, we add the terms $\mathcal{D}[L_{0,2j}](\rho)$ and $\mathcal{D}[L_{2,2j-1}](\rho)$ for $j=N$ to Section˜III.1, where the site label $i+2N\vcentcolon=i$. The jump operators $L_{\mu,j}$ are

Since all the jump operators commute with $S$, the Lindbladian $\mathcal{L}_{\mathcal{C}}$ has the $\mathbb{Z}_{2}^{S}$ symmetry, while it is easy to check that $\mathcal{D}[L_{1,j}]$ renders the $W$ symmetry weak. Thus, our parent Lindbladian has the $\mathbb{Z}_{2}^{\text{S}}\times\mathbb{Z}_{2}^{\text{W}}$ symmetry introduced in Section˜II.2. Each of the terms $L_{0,j}$, $L_{1,j}$, $L_{2,j}$ plays a distinct role in driving an arbitrary density matrix in the same symmetry sector as $\rho_{\mathcal{C}}$ to the steady state $\rho_{\mathcal{C}}$.

The jump operators have the following physical meanings. $L_{0,j}$ for even $j$ is designed to stabilize the domain-wall configurations. If the configuration is violated, we have $(\mathbbm{1}-Z_{j-1}X_{j}Z_{j+1})/2=+1$, and we fix the configuration by applying $X_{j+1}$. Note that this configuration can also be fixed by applying $X_{j-1}$, $Z_{j}Z_{j+2}$, $Z_{j}$, etc. However, applying $Z_{j}$ is forbidden as it breaks the strong symmetry $\mathbb{Z}_{2}^{S}$; we choose to apply $X_{j+1}$ because the resulting jump operator has the lowest weight. Now within the domain-wall configuration space, $L_{1,j}$ is designed to decohere the “off-diagonal” domain-wall configurations, leaving only $\outerproduct{\Psi_{\{z_{i}\}}}{\Psi_{\{z_{i}\}}}$, while $L_{2,j}$ makes the probability of getting all the configurations equal.

Note that $\rho_{\mathcal{C}}$ is a steady state for any values of the rates of the jump operators, but we will choose these rates to be equal. We have verified that changing the rates does not change the physics qualitatively.

Recall that the decohered cluster state is a mixed-state SPT state protected by a $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ symmetry, and the parent Lindbladian $\mathcal{L}_{\mathcal{C}}$ has the same $\mathbb{Z}_{2}^{S}\times\mathbb{Z}_{2}^{W}$ symmetry. The strong symmetry of the Lindbladian forces there to be at least one steady state in each $s_{\text{ket}}=s_{\text{bra}}^{*}$ physical strong symmetry sector. The argument is as follows. Because the Lindbladian is strongly symmetric, it conserves strong symmetry charge. That is, $S\rho=s_{\text{ket}}\rho$ and $\rho S=s_{\text{bra}}\rho$ implies

and similarly when acting from the bra side.

One can then consider the dynamics where one prepares an initial state with the symmetry charge $s_{\text{ket}}=s_{\text{bra}}^{*}$ and evolve it with the strongly symmetric Lindbladian. At an infinite time, such an initial state has to evolve into some state with the same strong symmetry charges, hence forcing the Lindbladian to have at least one steady state in the corresponding strong symmetry sector.

On the other hand, it is not guaranteed that a steady state will exist in the strong symmetry sectors $s_{\text{ket}}\neq s_{\text{bra}}^{*}$ or with a nontrivial weak charge. The above argument breaks down since one cannot prepare a physical initial state in the said symmetry sectors, as a state in the said symmetry sectors is traceless. More specifically, in the strong case, $\text{Tr}[\rho]=\text{Tr}[S\rho S^{\dagger}]=(s_{\text{ket}}s_{\text{bra}})\text{Tr}[\rho]$, which implies $s_{\text{ket}}s_{\text{bra}}=1$ or $\text{Tr}[\rho]=0$. In the weak case, $\text{Tr}[\rho]=\text{Tr}[W\rho W^{\dagger}]=w\text{Tr}[\rho]$, so $w=1$ or $\text{Tr}[\rho]=0$.

Since $\rho_{\mathcal{C}}$ is in the $s_{\text{ket}}=s_{\text{bra}}=+1$ sector, it is interesting to examine what is the state in the $s_{\text{ket}}=s_{\text{bra}}=-1$ sector. As we will show, the corresponding steady state in the $s_{\text{ket}}=s_{\text{bra}}=-1$ sector is $\rho_{-}\propto\sum_{j\in\text{even}}Z_{j}\rho_{\mathcal{C}}Z_{j}$.

In fact, we can partially solve and understand the parent Lindbladian $\mathcal{L}_{\mathcal{C}}$ by mapping the system to a dual model. This is akin to understanding the one-dimensional pure cluster state and its parent Hamiltonian by conjugating them via a depth-two circuit composed of CZ (controlled-Z) gates [42]. Recall that, for the pure cluster state defined in Eq.˜1,

where $\text{CZ}_{j,j+1}\vcentcolon=(1+Z_{j})/2+(1-Z_{j})Z_{j+1}/2$ is the controlled-Z two-qubit gate and $j+2N\!\vcentcolon=\!j$. The circuit is illustrated in Fig.˜1. Note that $\text{CZ}_{j,j+1}$ is invariant under $j\longleftrightarrow j+1$ and $U_{\text{CZ}}Z_{j}U_{\text{CZ}}^{\dagger}=Z_{j}$. Since $U_{\text{CZ}}X_{j}U_{\text{CZ}}^{\dagger}=Z_{j-1}X_{j}Z_{j+1}$, the operators $S$ and $W$ are invariant under the conjugation by $U_{\text{CZ}}$. In the following, we apply the $U_{\text{CZ}}$ circuit to map $\rho_{\mathcal{C}}$ to a tensor-product state $\tilde{\rho}_{\mathcal{C}}$ and $\mathcal{L}_{\mathcal{C}}$ to the Lindbladian $\tilde{\mathcal{L}}_{\mathcal{C}}$, allowing us to solve for the steady states and some of the eigenpairs (eigenvalues and corresponding eigenvectors) of $\mathcal{L}_{\mathcal{C}}$.

First, let us examine the state $\tilde{\rho}_{\mathcal{C}}=U_{\text{CZ}}\rho_{\mathcal{C}}U_{\text{CZ}}^{\dagger}$. Recall that $\rho_{\mathcal{C}}$ can be written as in Eq.˜3, which is an ensemble of the domain wall configurations $\ket{\Psi_{\{z_{i}\}}}$ given in Eq. (2) It is easy to see that

Summing over all the possible configurations of $z_{i}$ on the “Z”-qubits, we have

which is a tensor-product state with the same $\mathbb{Z}_{2}^{\text{S}}\times\mathbb{Z}_{2}^{\text{W}}$ symmetry realized by $S$ and $W$.

We obtain the jump operators defining $\tilde{\mathcal{L}}_{\mathcal{C}}$ by conjugating the original jump operators by $U_{\text{CZ}}$, giving us

where $j$ is on all the even sites for $\tilde{L}_{0,j}$ and on all the odd sites for $\tilde{L}_{1,j}$ and $\tilde{L}_{2,j}$.