We study steady-states of quantum Markovian processes whose evolution is described by local Lindbladians. We assume that the Lindbladian is gapped and satisfies quantum detailed balance with respect to a unique full-rank steady state *σ*. We show that under mild assumptions on the Lindbladian terms, which can be checked efficiently, the Lindbladian can be mapped to a local Hamiltonian on a doubled Hilbert space that has the same spectrum and a ground state that is the vectorization of *σ*^{1/2}. Consequently, we can use Hamiltonian complexity tools to study the steady states of such open systems. In particular, we show an area-law in the mutual information for the steady state of such 1D systems, together with a tensor-network representation that can be found efficiently.

## I. INTRODUCTION

Understanding the structure and physical properties of open many-body quantum systems is a major problem in condensed matter physics. Over the years, these systems have been studied using a plethora of methods from statistical physics, many-body quantum theory, and functional analysis. More recently, with the advent of quantum computation and quantum information, these systems have also been studied using techniques coming from quantum information. Coming from these research paradigms, one would typically want to understand the computational complexity of these systems, the type of entanglement and correlations that they can create, and whether or not they can be represented efficiently on a classical computer.

This line of research has been well-established in the context of *closed systems*. In particular, there are many results characterizing the complexity of ground states of local Hamiltonians defined on a lattice. For example, it has been shown that for several families of Hamiltonians it is QMA-hard to approximate the ground state energy,^{1,2} implying that the ground state of such systems does not posses an efficient classical description unless QMA = NP. On the other hand, it is generally believed that gapped local Hamiltonians on a *D*-dimensional lattice satisfy an area-law of entanglement entropy^{3} and can be well approximated by efficient tensor network states. This has been rigorously proven in 1D,^{4} with some partial results in higher dimension.^{5–11}

*σ*for which

*H*=

*∑*

_{i}

*h*

_{i}is a local Hamiltonian with ground state $\Omega $ and ground energy

*ϵ*

_{0}. Moreover, as described in Sec. II, the spectrum of $L$ is in the Re ≤ 0 part of the complex plane, and we can define the spectral gap of the Linbladian to be the largest non-zero real part of the spectrum. It is therefore tempting to try and use local Hamiltonian techniques to characterize

*σ*. However, a quick inspection reveals the main obstacle for such a simple plan to work: Whereas the Hamiltonian is a self-adjoint (Hermitian) operator, which therefore has a real spectrum with a set of orthonormal eigenstates, the same is not true for the Lindbladian; it is generally not a self-adjoint operator, and consequently its spectrum might be complex with non-orthogonal eigenoperators. To make $L$ self-adjoint, one might consider $L*L$, but this come at the price of losing locality. Alternatively, we might consider $12(L+L*)$, but it is not clear how the eigenstates of this operator are related to the eignstates of $L$.

A more sophisticated way of obtaining self-adjointness is by assuming that the Linbladian satisfies quantum detailed-balance.^{12,13} For such Linbladians there is a way of defining an inner product with respect to which $L*$ is self-adjoint^{13–16} (see Subsection II C). This inner product, however, can be highly non-local (with respect to the underlying tensor-product structure) and might deform the natural geometry of the Hilbert space significantly. It is therefore not clear how to use it to bound correlations and other measures of locality in the steady state.

Nevertheless, in this paper we show that the quantum detailed-balance condition provides a surprisingly simple map that takes a local Lindbladian to a self-adjoint *local* super-operartor (i.e., a super Hamiltonian), which is then mapped to a local Hamiltonian using vectorization. All this is done while maintaining a direct relation between the steady state of the former and the ground state of the latter. Consequently, many of the bounds and properties that were proved for ground states, easily transform to the steady states of detailed-balance Lindbladians. In particular, we show that under mild conditions, which can be efficiently checked, steady states of gapped 1D Lindbladian with detailed balance satisfy an area-law for the mutual information. Moreover, just as in the 1D area-law case for local Hamiltonians, these steady states are well-approximated by an efficient tensor network.

We conclude this section by noting that our mapping is not new; it has already been used before, e.g., in Refs. 15 and 17–20. Our main contribution is showing that this mapping results in a *local* Hamiltonian, rather than a general Hermitian operator, which enables us to take advantage of the local Hamiltonian machinery (see Subsection III A). It is important to note that the locality of Hamiltonian is non-trivial for general detailed balance local Lindbladian. For example, in Refs. 17 and 19, locality was achieved due to the further assumptions made in each work.

### A. Comparison with previous works

Several works studied the entanglement structure of steady states of open systems, see for example Refs. 18 and 21–23. Here we will concentrate on Refs. 18 and 21, which study problems that are very close to ours.

*D*-dimensional lattice with unique, full-rank steady state. Under the assumption of gapped, detailed-balanced Linbladian, they have shown exponential decay of correlations in the steady state. To prove an area law, the authors additionally assumed a very strong form of fast convergence to the steady state, known as system-size independent log-Sobolev constant (see Ref. 24 for a definition). Under this assumption, and using the Lieb-Robinson bounds for open systems,

^{25–27}they showed clustering of correlations in terms of the mutual information, and as a consequence, an area law for the mutual information. Specifically, for every region

*A*in the lattice, the mutual information is bounded by

*σ*

^{−1}‖ grows at least exponentially with system size, and might even grow doubly exponential with system size, in which case (1) is no longer an area-law.

*rapid mixing*. Roughly, it assumes that the restriction of the system to any region Λ on the lattice has a unique fixed point

*σ*

_{Λ}, to which it converges exponentially fast

*from any initial state*,

*c*,

*δ*,

*γ*are some region-independent constants. They manged to prove an area-law by assuming either that the Lindbladian is frustration free, or that the steady state is a pure state. Notice that the latter assumption addresses setups relevant to dissipative quantum state engineering,

^{28}although rapid mixing restricts it to low-entangled quantum states (see Ref. 28). Under these assumptions, using Lieb-Robinson bounds, they have managed to show the following area-law with logarithmic corrections

It is interesting to contrast these two results with several area-law results for ground states of gapped local Hamiltonians.^{4,11,29–31} In both cases, the proof follows the intuitive logic in which a fast convergence to the steady state (or ground state) can yield a bound on its entanglement. Indeed, given a region *A* in the lattice, we can prepare the system in a product state of this region with the rest of the system and then drive the system toward its steady state. If the convergence is quick and the underlying dynamic is local, not too much entanglement is created, which bounds the entanglement in the steady state.

There is, however, a highly non-trivial caveat in this program. It is not *a priori* clear that there exists a product state with a large overlap with the fixed point. If the overlap is exponentially small, it might take a long (polynomial) time for the dynamics to converge, even if the convergence is exponentially fast. This is the hard step in proving an area-law for ground states of gapped local Hamiltonians. In Hastings’ 1D area-law Proof,^{4} the initial state is $\rho A\u2a02\rho Ac$, where *ρ*_{A}, $\rho Ac$ are the reduced density matrices of the ground state on the regions *A*, *A*^{c}. Then a non-trivial overlap with $\rho A\u2a02\rho Ac$ is shown using an ingenious argument about the saturation of mutual information. In Refs. 11 and 29–31 it is done by constructing an approximate ground state projector (AGSP) using a low-degree polynomial of the Hamiltonian. If the Schmidt rank of this AGSP times its approximation error is smaller than unity, we are promised that there exists some product state $A\u2a02Ac$ with a large overlap with the groundstate.

In the open system area-law proof of Refs. 18 and 21 there is no parallel argument to lower-bound the overlap between the initial product state and the steady state. In addition, the convergence to the fixed point is always via the natural $etL$ map—which might not be the most efficient one (in terms of the amount of entanglement that is generated). In Ref. 18 a worse-case overlap is assumed, which leads to the log(‖*σ*^{−1}‖) factor in (3). (‖*σ*^{−1}‖ is the smallest possible overlap of full-supported *σ* with another state.) In Ref. 21, it is *assumed* that fast convergence is independent of the initial state. This assumption, together with additional local fixed-point uniqueness assumptions that are also made in that work, imply that local expectation values at the fixed point can be efficiently calculated by a classical computer, which might make it less interesting from a computational point of view. In that respect, we believe that our Lindbladian → Hamiltonian mapping paves the way for a more fine-grained analysis of the problem. Finally, we remark that the assumptions of a gap plus detailed-balance do *not* generally imply short mixing times. Indeed, in the East model^{32,33} there is a gap and yet the mixing time is linear. This separates our result from Refs. 18 and 21.

The structure of the paper is as follows. In Sec. II we introduce the notation and background of the paper. In Sec. III we state and prove our main results. In Sec. IV we describe two non-trivial models that satisfy the assumptions we make, and present explicit forms for their super-Hamiltonian. We explicitly derive resultant local Hamiltonians and show it is indeed a local (or exponentially local). We then use Hamiltonian complexity techniques from Refs. 34 and 35 to demonstrate a finite gap in those Hamiltonians (and correspondingly in the original $L$) for a specific example. In Appendix A we prove a lemma used in Theorem 3.2 regarding square root of sparse matrices. In Appendices B and C we prove that our examples indeed satisfy all the requirements and explicitly derive the super-Hamiltonian.

## II. BACKGROUND

### A. Setup

Throughout of this manuscript we use the big-O notation of computer science. If *n* is the asymptotic variable, then $X=OY$ means that there exists *C* > 0 such that for sufficiently large *n*, *X*(*n*) ≤ *C* · *Y*(*n*). On the other hand, $X=\Omega Y$ means that there exists a constant *c* > 0 such that *X*(*n*) ≥ *c* · *Y*(*n*) for sufficiently large *n*. Finally, $X=\Theta Y$ means that there exist constants *C* > 0, *c* > 0 such that *c* · *Y*(*N*) ≤ *X*(*n*) ≤ *C* · *Y*(*n*) for suffciently large *n*.

*d*-dimensional local Hilbert space (“qudits,” which could physically correspond to spins, fermions, or hard core bosons) that reside on a lattice Λ of

*n*sites and fixed spatial dimension (1D, 2D or 3D). The

*d*-dimensional local Hilbert space at site

*x*∈ Λ is denoted by $Hx$, so that the global Hilbert space is

*p*th Schatten norm of an operator

*X*by $\Vert X\Vert p=defTr(|X|p)1/p$. The

*p*= 1 case is the trace norm, commonly used to measure distance between density matrices. The

*p*= 2 case is the norm derived from the Hilbert-Schmidt inner-product. The

*p*= ∞ norm, defined by the maximal singular value of

*X*, is the operator norm, and is denoted in this paper by ‖

*X*‖.

Given a subset of the lattice *S* ⊆ Λ, we denote its local Hilbert space by $HS$, i.e., $HS=\u2a02x\u2208SHx$. An operator $O\u2208L(H)$ is *supported* on a subset *S* ⊆ Λ if it can be written as $O=OS\u2a021HSc$ where $OS\u2208L(HS)$ and $1HSc$ is the identity operator on the complementary Hilbert space.

Linear operators acting on the operators space $L(H)$ are called *superoperators*, and will usually be denoted by curly letters (e.g., $L,P$). As in the case of operators, we say a superoperator $B$ is supported on a subset *S* ⊆ Λ of the sites in the lattice if it can be written as $BS\u2a02ISc$, where $BS$ is a superoperator on $L(HS)$, and $ISc$ is the identity operation on the complementary operators space.

*self-adjoint*if $B=B*$.

### B. Open quantum systems

In this section we provide some basic definitions and results about Markovian open quantum systems. For a detailed introduction to this subject we refer the reader to Refs. 36 and 37.

*Lindbladian*(also known as a

*Liouvillian*) $L$ in the Schrödinger picture. Formally, this means that the quantum state describing the system evolves continuously by a family of

*completely positive trace preserving*(CPTP) maps ${Tt}t\u22650$, which are given by $Tt=defeLt$ so that

*dynamical semigroup*or

*Quantum Markovian Semigroup*due to the identity $Tt\u25e6Ts=Tt+s$. A necessary and sufficient condition for $L$ to be the generator of a semigroup is given by the following theorem (see, for example, Theorem 7.1 in Ref. 37):

*is a dynamical semigroup iff it can be written as*

*where*

*H*

*is a Hermitian operator and*

*L*

_{j}

*are*

*operators.*

The operator *H* is known as the Hamiltonian of the system, and it governs the coherent part of the evolution. The operators *L*_{j} are often called *jump operators*, and are responsible for the dissipative part of the evolution.

Given a Lindbladian $L$, the representation in (7) is not unique; for example, we can always change $H\u2192H+c1$ for $c1$ for $c\u2208R$ without changing $L$. In addition, $L$ will remain the same under the transformation $Lj\u2192Lj+cj1$ and $H\u2192H+i2\u2211j(cj*Lj\u2212cjLj\u2020)$. Therefore, we can assume without loss of generality that there is a representation of $L$ in which both *H* and *L*_{j} are traceless.

This condition, however, does not fully fix the jump operators, and there can be several traceless jump operators representations of the same Lindbladian. The following theorem, which is an adaptation of Proposition 7.4 in Ref. 37, shows how these representations are related.

**(Freedom in the representation of** $L$**, Proposition 7.4 in Ref. 37).** *Let* $L$ *be given by Eq. (7) with traceless* {*L*_{j}} *and* *H**. If it can also be written using traceless* ${Li\u2032}$ *and* *H*′*, then* *H* = *H*′ *and there exists a unitary matrix* *U* *such that* $Li\u2032=\u2211jUijLj$*, where the smaller set of jump operators is padded with zero* *operators.*

The following are two important properties of Lindbladians that will be used extensively in this work. First, we define the notion of locality of a Lindbladian (which can be generalized to a general super-operator).

**(***k***-body Lindbladians).** *We say that* $L$ *is a* *k**-body Lindbladian if it can be written as in (7) with jump operators* *L*_{j} *that are supported on at most* *k* *sites, and in addition the Hamiltonian* *H* *can be written as* *H* = *∑*_{i}*h*_{i}*, with every* *h*_{i} *also supported on at most* *k* *sites. We say that* $L$ *is a geometrically local* *k**-body Lindbladian if, in addition to being* *k**-body, every* *L*_{j} *and* *h*_{i} *are supported on neighboring lattice* *sites.*

Note that we are using the name “k-body” Lindbladian instead of “*k*-local:” as it is often done in the Hamiltonian complexity literature,^{1} the term *k*-local is reserved to local terms supported on at most *k* *qubits*. Here, we are allowing any constant local dimension *d*.

Next, we define the spectral gap of a Lindbladian.

**(Spectral gap of a Lindbladian).**

*The spectral gap of the Lindbladian is defined as the minimal real part of its non-zero eigenvalues*

The spectral gap of a Lindbladian controls the *asymptotic* convergence rate of the dynamics to a steady state,^{38} meaning that at *long enough times*, the distance to the steady state decays at rate lower bounded by the gap. However, it is important to notice that a finite gap by itself does not guarantee short-time convergence^{39} (see for example Refs. 32 and 33).

We conclude this section by listing few well-known facts about Lindbladians. We refer the reader to chapters 6 and 7 of Ref. 37 for proofs and details:

Fact 1: $L$ is an hermicity-preserving superoperator: $L(A)\u2020=L(A\u2020)$, and the same holds for $L*$.

Fact 2: There is always at least one quantum state

*σ*that satisfies $L(\sigma )=0$, namely*σ*is a fixed point of the time-evolution. We refer to it as a*steady**state.*Fact 3: $Tt$ is a contractive map and consequently, $L$ has only non-positive real parts in its spectrum (Proposition 6.1 in Ref. 37).

### C. Quantum detailed-balance

In this work, we follow Refs. 13, 14, and 16 in defining the detailed-balance condition for the Lindbladian system. We begin by describing classical detailed-balance, and then use it to define the corresponding quantum condition.

Classically, let $P\u2208Mn$ be the transition matrix of a Markov chain over the discrete state of states {1, 2, …, *n*}, i.e., $Pij=defProb(j\u2192i|j)$, and let *π* denote a probability distribution on these states. Then *P* is said to satisfy the detailed-balance condition with respect to a fully supported *π* (i.e., *π*_{i} > 0 for all *i* ∈ [*n*]) if the probability of observing a *i* → *j* transition is identical to the probability of observing a *j* → *i* transition, when the system state is described by *π*. Mathematically, this means *P*_{ij}*π*_{j} = *P*_{ji}*π*_{i}. This condition implies that *π* is a steady state of the Markov chain, however, the converse is not always true.^{40}

_{π}to be the diagonal matrix $(\Gamma \pi )ij=\pi i\delta ij$, the detailed-balance condition can also be written as the matrix equality:

_{π}is a positive definite matrix, the above condition is equivalent to the condition of $\Gamma \pi \u22121/2P\Gamma \pi 1/2$ being symmetric:

*π*→

*σ*for some reference quantum state

*σ*. Just as in the classical case, we demand that

*σ*is invertible, i.e., it has no vanishing eigenvalues. The quantum analog of Eq. (9) should be an equation over superoperators that act on quantum density operators. While

*P*is replaced by $L$, Γ

_{π}should be replaced by a superoperator that multiplies an input state by the quantum state

*σ*. But as

*σ*does not commute with all quantum states (unless it is the completely mixed state), there are several ways to define this multiplication. In particular, for every

*s*∈ [0, 1], we might define the “multiplication by

*σ*” superoperator (Here we do not use a curly letter to denote the superoperator Γ

_{s}in order to be consistent with previous works.)

_{s}is self adjoint: $\Gamma s=\Gamma s*$. Moreover, Γ

_{s}is invertible, and for every $x\u2208R$, we have $\Gamma sx(A)=\sigma x(1\u2212s)A\sigma xs$.

With this notation, the quantum detailed balance (QDB) condition is defined as the following generalization of the classical condition (9):

**(Quantum detailed balance).**

*A Lindbladian satisfies quantum detailed-balance with respect to some invertible (full-rank) state*$\sigma \u2208L(H)$

*and*

*s*∈ [0, 1]

*if*

*where*Γ

_{s}

*is the superoperator defined in (11).*

We note that not every steady-state of a Lindbladian defines a superoperator Γ_{s} with respect to which the Lindbladian obeys detailed-balance.^{41}

As in the classical case, the quantum detailed-balance condition can be formulated in a few equivalent ways:

*Given a Lindbladian* $L$*, an invertible state* $\sigma \u2208L(H)$*, and* *s* ∈ [0, 1]*, the following conditions are equivalent:*

*(QDB1)*$L$*satisfies quantum detailed balance with respect to**σ**for some**s*.*(QDB2) The superoperator*$\Gamma s\u22121/2\u25e6L\u25e6\Gamma s1/2$*is self-adjoint, namely*$\Gamma s\u22121/2\u25e6L\u25e6\Gamma s1/2=\Gamma s1/2\u25e6L*\u25e6\Gamma s\u22121/2$.*(QDB3)*$L*$*is self-adjoint with respect to the inner-product defined by*(13)$A,Bs=defTr(A\u2020\u22c5\Gamma s(B))=Tr(A\u2020\u22c5\sigma 1\u2212sB\sigma s).$

^{1/2}, and (QDB3) is equivalent to (QDB1) by

The *s* = 1 is commonly known as the Gelfand-Naimark-Segal (GNS) case, and its inner product $A,B1=Tr(\sigma A\u2020B)$ is often called the GNS inner-product. Similarly, the *s* = 1/2 case is called the Kubo-Martin-Schwinger (KMS) case, with $A,B1/2$ known as the KMS inner-product.

It was shown in Ref. 16 (see Lemmas 2.5 and 2.8 therein) that any superoperator that is self adjoint with respect to the inner product defined by some *s* ∈ [0, 1/2) ∪ (1/2, 1] is self adjoint with respect to the inner product defined by all *s*′ ∈ [0, 1], including *s*′ = 1/2. Therefore by QDB3, if $L$ satisfies the detailed-balance condition for some *s* ∈ [0, 1/2) ∪ (1/2, 1], then it satisfies it for all other *s* ∈ [0, 1].

*σ*, then

*σ*is automatically a steady state, since $\Gamma s\u22121(\sigma )=1$ and therefore

### D. The canonical form

The QDB condition has well-known implications to the structure of the Lindbladian $L$. In this section we describe some of the central consequences of this condition, and in particular the so-called *canonical form*, which will be used later. None of the results in this section are new, as they already appeared in several works (see, for example, Refs. 13, 16, and 44). Nevertheless, we repeat some of the easy proofs for sake of completeness.

Our starting point is the *modular superoperator*, which is central for constructing a canonical representation of detail-balanced Lindbladians.

**(The modular superoperator).**

*Given an invertible state*

*σ*

*, its associated modular superoperator is defined by*

*We note that we are using a greek letter*Δ

*to denote the modular superoperator instead of curly letter. This is done for being consistent with the notations of Refs. 16 and 20.*

Below are few properties of Δ_{σ}, which follow almost directly from its definition.

*The modular superoperator* Δ_{σ} *has the following properties:*

$\Delta \sigma \u22121(A)=\sigma \u22121A\sigma $

*and*$\Delta \sigma (A)\u2020=\Delta \sigma \u22121(A\u2020)$.*Self-adjointness:*$\Delta \sigma *=\Delta \sigma $.*Positivity:*$A,\Delta \sigma (A)>0$*for all non-zero operators**A*.

*A*≠ 0, then by the invertability of

*σ*it follows that also

*σ*

^{1/2}

*Aσ*

^{−1/2}≠ 0, hence the RHS above is positive.■

_{σ}is fully diagonalizable by an orthonormal eigenbasis of operators {

*S*

_{α}} and positive eigenvalues ${e\u2212\omega \alpha}$, where

*α*= 0, 1, 2, …,

*N*

^{2}− 1 is a running index. As $\Delta \sigma (1)=1$, we can fix $S0=1N1$,

*ω*

_{0}= 0, and conclude that $Tr(S\alpha )=NS0,S\alpha =0$ for every

*α*> 0. Finally, from property (1) we find that

_{α}with eigenvalue $e\omega \alpha $. All of these properties are summarized in the following corollary, which defines the notion of a

*modular*

*basis*.

**(Modular basis).** *Given an invertible state* *σ**, the modular superoperator* Δ_{σ} *has an orthonormal diagonalizing basis* {*S*_{α}} *with* *α* = 0, 1, …, *N*^{2} − 1 *and the following properties:*

$S\alpha ,S\beta =\delta \alpha \beta $

$S0=1N1$

*and*Tr(*S*_{α}) = 0*for**α*> 0.$\Delta \sigma (S\alpha )=e\u2212\omega \alpha S\alpha $

*and*$\Delta \sigma (S\alpha \u2020)=e\omega \alpha S\alpha \u2020$.*For every**α**there exists**α*′*such that*$S\alpha \u2020=S\alpha \u2032$.

*The basis* {*S*_{α}} *is called a modular* *basis.*

Being a diagonalizing basis, the modular basis is unique up to unitary transformations within every eigenspace. The numbers *ω*_{α}, which determine the eigenvalues of the modular superoperator, are called *Bohr frequencies*. We say that an operator *O* has well-defined Bohr frequency *ω* if Δ_{σ}(*O*) = *e*^{−ω}*O*, i.e., it belongs to the eigenspace of Δ_{σ} with eigenvalue *e*^{−ω}. For example, it is evident that *σ* itself (or any other state that commutes with it) has a well-defined Bohr frequency *ω* = 0 since Δ_{σ}(*σ*) = *σ*.

*σ*. To see this, we use an explicit construction of a modular basis. Given the spectral decomposition $\sigma =\u2211ie\u2212Eiii$, we define the operators $Oij=defij$ with

*i*,

*j*∈ 1, …,

*N*, and note that they form an orthonormal diagonalizing basis of Δ

_{σ}:

Finally, Lindbladians that satisfy the QDB condition for *s* ∈ [0, 1/2) ∪ (1/2, 1] with respect to an invertible state *σ* can be written in a particular canonical way. This was first proved in Ref. 13 under slightly different conditions. Here we will follow Ref. 16, and use an adapted form of Theorem 3.1 from that reference:

**(Canonical form of QDB Lindbladians, adapted from Theorem 3.1 in Ref. 16).**

*A Lindbladian*$L$

*satisfies quantum detailed-balance with respect to an invertiable state*

*σ*

*and*

*s*∈ [0, 1/2) ∪ (1/2, 1]

*if and only if it can be written as*

*where*

*ω*

_{α}

*are Bohr frequencies, the jump operators*{

*S*

_{α}}

*are taken from a modular basis, and*{

*γ*

_{α}}

*are positive weights that satisfy (*Recall that in accordance with the modular basis definition (see Corollary 2.9), for every index

*α*, there exists an index

*α*′ such that $S\alpha \u2020=S\alpha \u2032$ and

*ω*

_{α}= −

*ω*

_{α′}.)

*γ*

_{α}=

*γ*

_{α′}

*, and in particular the set of indices*

*I*

*contains*

*α*′

*for every*

*α*∈

*I*.

We end this section with a few remarks.

In the original text of Ref. 16, the formula for $L$ is given in the Heisenberg picture, which is the formula for $L*$ in our notation. Additionally, the

*γ*_{α}weights are missing, as they are instead absorbed into the*S*_{α}. The normalization condition $S\alpha ,S\beta =\delta \alpha \beta $, however, remained unchanged, which we believe is a mistake.In Eq. (15) we sum over strictly positive

*γ*_{α}and ignore the vanishing weights. In Ref. 16 [Eq. (3.3)] the authors considered the vanishing coefficients as well in the summation.It follows from the canonical representation that if $L$ satisfies the QDB condition, it is purely dissipative, i.e., its Hamiltonian is vanishing. In the literature, QDB condition is often referred to the dissipator part of the Lindbladian.

Theorem 2.10 applies to any Lindbladian defined on a finite dimensional Hilbert-space regardless of the many-body structure of the underlying Hilbert space, which can describe spins, fermionic, or hard-core bosonic systems.

A physical example of a Lindbladian that satisfies the QDB condition (for

*s*= 1) is the Davies generator of some Hamiltonian*H*.^{45}It is the principle example of a semigroup whose unique fixed point is a Gibbs state (i.e., thermal state), and is often referred to as a thermal semigroup.^{19,24,46}

## III. MAIN RESULTS

In this section we give precise statements of our main results about the mapping of detailed-balanced local Lindbladians to local Hamiltonians (Subsection III A), and discuss their application to the complexity of the steady states of these systems (Subsection III B). The proofs of the main results are given in Subsection III C.

### A. Mapping detailed-balanced Linbaldians to local Hamiltonians

*k*-body Lindbladian $L$ defined on a finite

*D*dimensional lattice Λ of qudits with local dimension

*d*. We assume that $L$ satisfies the QDB condition in Definition 2.5 with respect to some

*s*∈ [0, 1/2) ∪ (1/2, 1] and a unique, full-rank steady state

*σ*. As $L$ is detailed-balanced, it does not have a Hamiltonian part, and therefore it can be written as

*k*-body traceless jump operators

*L*

_{j}. We also assume that $L$ has a spectral gap

*γ*> 0. Note that since $L$ obeys detailed-balance, it has a real spectrum, and $gap(L)=\gamma $.

*σ*to a local Hamiltonian problem, where they can be analyzed using plethora of well-established tools.

^{4,30,34,47,48}This is achieved by mapping the Lindbladian to the super-operator

*super-Hamiltonian*, and study its vectorization as a local Hamiltonian. (To obtain a physical interpretation for the super Hamiltonian, one can relate its imaginary time evolution to the Lindblad evolution by similarity transformation, namely, $e\u2212Ht=\Gamma s\u22121/2\u25e6eLt\u25e6\Gamma s1/2$.)

Since $H$ and $L$ are related by a similarity transformation, they share the same spectrum (up to an overall global minus sign), and therefore also $H$ has a spectral gap *γ* > 0. Moreover, it is easy to see that an eigenoperator *A* of $L$ maps to an eigenoperator $\Gamma s\u22121/2(A)=\sigma \u2212(1\u2212s)/2A\sigma \u2212s/2$ of $H$, and in particular the steady state *σ* maps to $\sigma $ which is in the kernel of $H$. We remark that the superoperator $\Gamma s\u22121/2\u25e6L\u25e6\Gamma s1/2$ and its steady state $\sigma $ were already studied in Refs. 15, 17, and 18 as tools for relating the *χ*^{2} decay constant to the spectral gap of a QDB Lindbladian, and in Ref. 19 for demonstrating strong clustering of information using the detectability lemma. While the locality of the super-Hamiltonian was not addressed in Refs. 15 and 18, it was used in Refs. 17 and 19. However, in these papers, locality was automatically achieved due to the steady state being a Gibbs state of a commuting local Hamiltonian, or the canonical (Davies) form having local jump operators. Our work is devoted to proving this in the general case.

We are left with the task of showing that $H$ is geometrically local, or at least local with decaying interactions. We will prove this under two possible assumptions. In Theorem 3.1 we assume a certain linear-independence conditions on the jump operators, and consequently find that $H$ is a *k*-body geometrically local. In Theorem 3.2 we expand the jump operators in terms of a local orthonormal basis, and show that if the coefficient matrix in that basis is gapped, then $H$ is 2*k*-body local with coefficients that decay exponentially with the lattice distance.

*Let*{

*L*

_{j}}

*be a set of linearly independent and normalized jump operators such that for every index*

*j*

*, there exists an index*

*π*(

*j*)

*such that*$Lj\u2020=L\pi (j)$

*. Assume that*$L$

*satisfies the QDB condition (Definition 2.5) for*

*s*∈ [0, 1/2) ∪ (1/2, 1]

*and is given by*

*for some positive coefficients*

*c*

_{j}> 0

*. Then the super-Hamiltonian (17) is given by*

*Consequently, if*$L$

*is*

*k*

*-body and geometrically local, then so is*$H$.

*P*

_{a}}, which we assume to be Hermitian and

*k*-body geometrically local:

*d*= 2, we can take {

*P*

_{a}} to be products of

*k*Paulis on neighboring sites. Substituting Eq. (20) in Eq. (16) yields

*Let*$L$

*be a Lindbladinan given by Eq. (21) with*{

*P*

_{a}}

*being a set of orthonormal hermitian operators, and assume that*$L$

*satisfies the QDB condition (Definition 2.5) with respect to*

*s*∈ [0, 1/2) ∪ (1/2, 1]

*. Then the super-Hamiltonian (17) is given by*

*where*

*C**

*is the complex conjugate of*

*C*.

We note as a side-product of the proof, *C*, *C** commute, and as they are both non-negative matrices, the square root $(C\u22c5C*)1/2$ is well-defined. As a corollary, we achieve a sufficient condition for quantum detailed balance, which can be easily checked with a polynomial computation *without the knowledge of the steady state* *σ*. Let us formally state this result:

**(A necessary condition for quantum detailed-balance).** *Let* $L$ *be a Lindbladinan given by Eq. (21) with* {*P*_{a}} *being a set of orthonormal hermitian operators, and assume that* $L$ *satisfies the QDB condition (Definition 2.5) with respect to* *s* ∈ [0, 1/2) ∪ (1/2, 1]*. Then the matrix* *C* *from Eq. (21) commutes with its complex conjugate* *C**.

Assuming $L$ is *k*-body and geometrically local, the matrix *C* is sparse: *C*_{ab} ≠ 0 only for *a*, *b* for which *P*_{a} and *P*_{b} appear in the expansion of the same jump operator *L*_{j}, and therefore their joint support is at most gemotrically *k*-local. This means that the *∑*_{a,b}*C*_{ab}{*P*_{b}*P*_{a}, *ρ*} term in Eq. (22) is geometrically-local *k*-body. However, the geometrical locality of the first term $\u2211a,b(C\u22c5C*)ab1/2Pa\rho Pb$ is not so clear, as it might mix *P*_{a} and *P*_{b} with distant supports (which might lead to a long-range $H$). In the following lemma, we show that as long as the smallest non-zero eigenvalue of *C* · *C** is Ω(1), the coefficient $(C\u22c5C*)ab1/2$ decays exponentially with the distance between the support of *P*_{a}, *P*_{b}, making the terms of $H$ decay exponentially with distance.

*Let*

*λ*

_{min}> 0

*be the smallest non-zero eigenvalue of*

*C*·

*C**

*, and assume that*|

*C*

_{ab}| ≤

*J*

*for every*

*a*,

*b*

*. Finally, let*|

*a*−

*b*|

*denote the lattice distance between the supports of*

*P*

_{a},

*P*

_{b}

*. Then*

*where*

*c*

_{1},

*c*

_{2}

*are constants that depend only on the geometry of the lattice and*

*k*.

The proof of this lemma is given in Appendix A.

*Let*$L$

*be a geometrically local*

*k*

*-body Lindbladian given by Eq. (21) that satisfies the QDB condition with respect to a full rank state and*

*s*∈ [0, 1/2) ∪ (1/2, 1]

*. Let*

*λ*

_{min}

*be the smallest non-zero eigenvalue of*

*C*·

*C**

*, and assume that*|

*C*

_{ab}| ≤

*J*

*for all*(

*a*,

*b*)

*pairs. Then the super-Hamiltonian in Theorem 3.2 is of the form*

*where*$Hab$

*is a*2

*k*

*-local super operator that acts non-trivially on*Supp(

*P*

_{a}) ∪ Supp(

*P*

_{b})

*, with an exponentially decaying interaction strength*$\Vert Hab\Vert =Je\u2212O(|a\u2212b|\lambda min/J2)$.

We conclude this section with two remarks:

Related to Corollary 3.3, also the linear independence of {

*L*_{j}} from Theorem 3.1, as well as the minimal non-vanishing eigenvalue of*C*·*C** from Theorem 3.2, can be checked and calculated efficiently without any prior knowledge of the steady state or the gap. This might be beneficial in numerical applications of these mappings, as the formula for $H$ is explicitly given in terms of the jump operators and*C*.Theorems 3.1 and 3.2 are valid for any Lindbladians defined on a finite-dimensional algebra of operators, as they do not use any commutativity or locality properties of the jump operators or basis. Therefore, these theorems apply to fermionic/hard-core bosonic systems as well.

### B. Application to the complexity of steady states of QDB Lindbladians

The mapping $L\u21a6H$ allows us to easily import results from the Hamiltonian realm into the Lindbladian realm. Here, we describe some of the main results that can be imported and discuss the underlying techniques. We begin by describing the vectorization mapping that allows us to map super-Hamiltonians and density operators to Hamiltonians and vectors, respectively.

#### 1. The vectorization map

*vectorization*, which is the well-known isomorphism $L(H)\u21a6H\u2a02H$ (see, for example, Ref. 49). We will denote this mapping by $X\u2192X$, where $X\u2208L(H)$ is an operator and $X\u2208H\u2a02H$ is a vector. For the sake of completeness, we explicitly define it here and list some of its main properties. The map is defined first on the standard basis elements by

*A*,

*B*) = Tr(

*A*

^{†}

*B*) = ⟨⟨

*A*|

*B*⟩⟩. It can also be extended naturally to many-body setup where we have a lattice Λ and the global space is $H=\u2a02x\u2208\Lambda Hx$. In such case, we view $H\u2a02H$ as composite system in which at every site

*x*∈ Λ, there are

*two*copies of the Hilbert space $Hx$: the original $Hx$ and a fictitious $Hx$, so that $H\u2a02H=\u2a02x\u2208\Lambda (Hx\u2a02Hx)$ as described in Fig. 1. With this definition, the notion of locality in $H$ can be naturally mapped to locality in $H\u2a02H$. This point of view is very intuitive when considering the vectorization $|\sigma 1/2\u27e9\u27e9$ of a quantum state

*σ*. First note that $|\sigma 1/2\u27e9\u27e9$ is a normalized vector since by the preservation of inner product,

*A*on $H$,

*A*with respect to

*σ*is equal to the expectation value of $A\u2a021$ with respect to $|\sigma 1/2\u27e9\u27e9$. This implies that $|\sigma 1/2\u27e9\u27e9$ is a

*purification*of

*σ*

_{fict}denotes the tracing over the fictitious $Hx$ Hilbert spaces. The above observations can be summarized in the following easy lemma, which relates the entanglement structure of $|\sigma 1/2\u27e9\u27e9$ to that of

*σ*.

*Let* *σ* *be a many-body quantum state on a* *D**-dimensional lattice* Λ*, let* $|\sigma 1/2\u27e9\u27e9$ *be the vectorization of* *σ*^{1/2}*, and define* $\rho =def|\sigma 1/2\u232a\u232a\u2329\u2329\sigma 1/2|$*. Then:*

*For any bi-partition*Λ =*A*∪*B**,*(24)$I(A:B)\sigma \u22642S(A)\rho ,$*where**I*(*A*:*B*)_{σ}*is the mutual information between**A*,*B**in the state**σ**, and**S*(*A*)_{ρ}*is the entanglement entropy of region**A**in the composite system*$H\u2a02H$*(where for every**x*∈*A**we include both the original system and its fictitious partner) with respect to the state*$|\sigma 1/2\u27e9\u27e9$*.**S*(*A*)_{ρ}*is also known as the operator-space entanglement entropy of*$\sigma $*(see Refs. 50 and 51).**If*Λ*is 1D and there exists an MPS*$\psi D$*on the composite system with bond dimension**D**such that*$\Vert |\sigma 1/2\u27e9\u27e9\u2212\psi D\Vert \u2264\u03f5$*, then*$\Psi D2=defTrfict\psi D\psi D$*can be described by an MPO with bond dimension**D*^{2}*and*$\Vert \sigma \u2212\Psi D2\Vert 1\u22642\u03f5$*. A similar relation exists also for higher dimensions (replacing MPS with, say,**PEPS).*

- By definition,
*I*(*A*:*B*)_{ρ}=*S*(*A*)_{ρ}+*S*(*B*)_{ρ}−*S*(*AB*)_{ρ}, and as*ρ*is a pure state and*A*,*B*a bi-partition of the system,*S*(*A*)_{ρ}=*S*(*B*)_{ρ}and*S*(*AB*)_{ρ}= 0. Therefore,*I*(*A*:*B*)_{ρ}= 2*S*(*A*)_{ρ}. As*σ*= Tr_{fict}*ρ*, it follows from the monotonicity of the relative entropy that$I(A:B)\sigma =S(\sigma \Vert \sigma A\u2a02\sigma B)\u2264S(\rho \Vert \rho A\u2a02\rho B)=I(A:B)\rho =2S(A)\rho .$ - First note thatand therefore $\psi D\sigma 1/2\u22651\u2212\u03f52/2$ and$\u03f5\u2265\Vert \sigma 1/2\u2212\psi D\Vert =2(1\u2212Re\u2329\psi D\sigma 1/2)\u22652(1\u2212|\u2329\psi D\sigma 1/2|)$Therefore, by the monotonicity of the trace distance,$\Vert \rho \u2212\psi D\psi D\Vert 1=\Vert \sigma 1/2\u232a\u232a\u2329\u2329\sigma 1/2\u2212\psi D\psi D\Vert 1=21\u2212|\sigma 1/2\psi D\u232a|2\u22642\u03f5.$$\Vert \sigma \u2212\Psi D2\Vert 1=\Vert Trfict\rho \u2212Trfict\psi D\psi D\Vert 1\u22642\u03f5.$

*D*

^{2}. This can be understood from inspecting Fig. 2, which shows how $\Psi D2$ is obtained by contracting the fictitious legs in the TN that describes $\rho =\sigma 1/2\u232a\u232a\u2329\u2329\sigma 1/2$.■

*H*. It is easy to verify that if $F$ is the superoperator $F(X)=defA\u22c5X\u22c5B$, then its vectorization is the operator

*F*=

*A*⨂

*B*

^{T}. Using this formula it is clear that a

*k*-body super-operator maps to a 2

*k*-body operator. For example, the super Hamiltonian (19) from Theorem 3.1 is mapped to the Hamiltonian

#### 2. Area laws for steady states

In this subsection we use the notation $O\u0303(X)$ for $OX\u22c5poly(log\u2061X)$. Our first result will be an area law for 1D Lindbladians that satisfy the requirements of Theorem 3.1. These map to geometrically local 1D super Hamiltonians, for which we will use the following result:

**(Taken from Refs. 30 and 47).** *Let* Λ *be a 1D lattice of sites with local dimension* *d**. Let* *H* = *∑*_{i}*h*_{i} *be a nearest neighbors Hamiltonian with a spectral gap* *γ* > 0 *and a unique ground state* $\Omega $*. Assume, in addition, that* ‖*h*_{i}‖ ≤ *J* *for every* *i**, where* *J* *is some energy scale. Then the entanglement entropy of* $\Omega $ *with respect to any bi-partition satisfies* $SE=O\u0303log3(d)\gamma /J$*. Moreover, there is a matrix product state (MPS)* $\psi D$ *of sublinear bond dimension* $D=eO\u0303(log3/4\u2061n/(\gamma /J)1/4)$ *such that* $\Vert \Omega \u2212\psi D\Vert \u22641poly(n)$*, and such MPS can be found efficiently on a classical* *computer.*

Using the above theorem together with Lemma 3.6 and Theorem 3.1, we immediately obtain the following corollary.

*Let* $L$ *be a geometrically local two-body Lindbladian defined on a 1D lattice of local dimension* *d* *that satisfies the requirements of Theorem 3.1. Assume in addition that* $L$ *has a gap* *γ* > 0*, a unique steady state, and that* |*c*_{j}| ≤ *J* *for all* *j**. Then:*

- $\sigma $
*satisfies an area law for the operator-space entanglement entropy,*Λ =^{50}that is, for any cut in the 1D lattice into*L*∪*R**the entanglement entropy of*$\sigma $*between the two parts satisfies*(25)$S(L)\rho =O\u0303log3(d)\gamma /J.$ *σ**satisfies an area-law for the mutual information. Specifically, for any cut in the 1D lattice, the mutual information between the two sides satisfies*(26)$I(L:R)\sigma =O\u0303log3(d)\gamma /J.$*There is a matrix product operator (MPO)*Ψ_{D}*with bond dimension*$D=eO\u0303(log3/4\u2061n/(\gamma /J)1/4)$*such that*$\Vert \sigma \u2212\Psi D\Vert 1\u22641poly(n)$*. Moreover, a similar approximating MPO for**σ**can be found efficiently on a classical**computer.*

For the next result, we will use the main result of Ref. 31 adapted to the two-local settings with exponentially decaying interactions.

**(Taken from Ref. 31).**

*Let*Λ

*a 1D lattice of sites with local dimension*

*d*.

*Let*

*H*=

*∑*

_{i,j}

*h*

_{ij}+

*∑*

_{i}

*h*

_{i}

*be a two-body Hamiltonian defined on*Λ

*, and suppose that there exist a constant*

*J*

*such that*$\Vert hij\Vert \u2264Jrij\alpha ,\Vert hi\Vert \u2264J$

*, and that*

*H*

*has a spectral gap*

*γ*> 0

*and a unique ground state*$\Omega $

*. Then the entanglement entropy of*$\Omega $

*with respect to any cut in the 1D grid satisfies*

*Moreover, there is a MPS*$\psi D$

*of bond dimension*$D=eOlog5/2(n)$

*such that*$\Vert \Omega \u2212\psi D\Vert \u22641poly(n)$.

It should be noted that Theorem 3.9 can be restated for 1D *k*-local Hamiltonians and still give the same entropy bound — see Appendix B in Ref. 31. Using the above theorem, together with Lemma 3.6 and Corollary 3.5, we can choose an appropriate *B* for which $\Vert Hab\Vert \u2264B/|a\u2212b|\alpha $ for any *P*_{a}, *P*_{b} for some *α* > 4. This implies the following corollary:

*Let* $L$ *be a geometrically local two-body Lindbladian defined on a* 1*D* *lattice that satisfies the requirements of Theorem 3.2, with a spectral gap* *γ* > 0 *and a steady state* *σ**. Assume also that the smallest non-vanishing eigenvalue of* *C* · *C** *is* $\lambda min=\Omega 1$*. Then*

- $\sigma $
*satisfies an area law for the operator-space entanglement entropy.*Λ =^{50}That is, for any cut in the 1D lattice into*L*∪*R**the entanglement entropy of*$\sigma $*between the two parts satisfies*(27)$S(L)\rho =O\u0303log2\u2061dlog\u2061d\gamma /J2.$ *σ**satisfies an area-law for the mutual information. Specifically, for any cut in the 1D lattice, the mutual information between the two sides satisfies*(28)$I(L:R)\sigma =O\u0303log2\u2061dlog\u2061d\gamma /J2.$*There is a matrix product operator (MPO)*Ψ_{D}*with bond dimension*$D=eOlog5/2(n)$*such that*$\Vert \sigma \u2212\Psi D\Vert 1\u22641poly(n)$.

We finish this part by noting that there are many more applications of the theorems in Subsection III A which we did not address. For example:

One can import other ground state area-law results to the Linbladian settings, including results for 2D and higher dimensions such as those given in Refs. 5, 7–9, and 11. For that purpose, one should check that the conditions required in the references above are fulfilled for the super-Hamiltonian. For instance, suppose that $L$ is a 2D Lindbladian given in a canonical form with local jump operators, and further assume it is locally gapped (in the sense discussed in Ref. 11). Then the super-Hamiltonian inherits the same locality structure, frustration-freeness and local gap of the original Lindbladian. The area law for the steady state is automatically achieved using the result of Ref. 11. An example to such model would be the 2D generalization of the toy-model described in Subsection IV A.

One can use the results in Refs. 48 and 52 to show exponential decay of correlations for $\sigma $, which implies the same for

*σ*using Eq. (23). This was already shown in Ref. 18 (see Theorem 9 in the paper), so we omit the details. Notice that our result are implied for any such Lindbladian without assuming local uniqueness (see*regular*Lindbladians in Ref. 18).One can demonstrate a gap in the Lindbladian by demonstrating a gap in the super-Hamiltonian instead. As the former task is somewhat challenging, the latter has been discussed more frequently in the literature. Below we will exemplify this by using the finite-size criteria from Ref. 34 to demonstrate a gap for a family of Lindbladians (see the proof of Claim 4.1, bullet 4 for details). We remark that this type of work has been also done in Ref. 17.

### C. Proofs of Theorems 3.1 and 3.2

In this section we give the proofs of our main results, Theorems 3.1 and 3.2. The two proofs follow the same outline: Examining the continuous family of superoperators $Lx=def\Gamma s\u2212x\u25e6L\u25e6\Gamma sx$, we would like to show that it obeys some notion of locality at *x* = 1/2. In fact, we will show that it is local for all $x\u2208R$. By definition, $L0=L$, and by the QDB condition (12), $L1=\Gamma s\u22121\u25e6L\u25e6\Gamma s=L*$, both of which are local. To prove locality for all other $x\u2208R$ we use the modular basis (see Corollary 2.9). We will show that $Lx$ remains diagonal in the modular basis for *every* *x*. Then, we will use Theorem 2.2 to establish a connection between the local representation of $L$ and the modular basis. Finally, we will employ the fact that $L1$ is local to argue that it must remain local for all $x\u2208R$.

Let us then begin by proving that $Lx$ remains diagonal in the modular basis.

*(Adaptation of Lemma 7 from Ref. 20). Let*$L$

*satisfy the quantum detailed-balance condition with respect to*

*s*∈ [0, 1/2) ∪ (1/2, 1]

*. Then for every*$x\u2208R$

*, the superoperator*$\Gamma s\u2212x\u25e6L\u25e6\Gamma sx$

*is diagonal in the modular basis, and is given by*

*Note:**the above formula is identical to the formula for*$L$

*in Theorem 2.10, except for the*$ex\omega \alpha $

*factor in front of the*$S\alpha AS\alpha \u2020$

*term.*

*t*,

With Lemma 3.11 at hand, we turn to the Proof of Theorem 3.1.

#### 1. Proof of Theorem 3.1

*U*

_{α,j}that connects these two bases, i.e.,

*L*

_{j}basis as

*B*

*B*

^{x}is diagonal, for at least one $x\u0303\u22600$, which will then imply that it also diagonal for

*all*$x\u2208R$. This is done using the QDB condition (12), which will let show diagonality for $x\u0303=1$. The QDB condition is equivalent to

*π*(

*j*), $L\pi (j)=Lj\u2020$. Let us now use the assumption that the jump operators {

*L*

_{j}} are linearly independent. This implies that the coefficients of $LjALj\u2032\u2020$ on both sides of the equation should be identical, and therefore,

*c*

_{j}> 0, we conclude that

*x*= 1/2 concludes the proof.■

#### 2. Proof of Theorem 3.2

*S*

_{α}that appear in $L$ in (15) (those with

*γ*

_{α}> 0) can be unitarily expressed by the orthonormal basis {

*P*

_{a}}. Indeed, starting from the jump operators representation in Eq. (16), we conclude from Theorem 2.2 that whenever

*γ*

_{α}> 0,

*S*

_{α}can be written in terms of the jump operators

*L*

_{j}. But as

*L*

_{j}can be expanded in terms of the

*local*{

*P*

_{a}} operators, it follows that this also holds for

*S*

_{α}as well. Finally, by the orthonormality of both sets of operators, we conclude that they are unitarily related: There exists a unitary

*U*

_{aα}, where

*α*runs over all the indices

*α*for which

*γ*

_{α}> 0, such that

*C*is the coefficient matrix of the expansion of $L$ in terms of the {

*P*

_{a}} operators [Eq. (21)].

*P*

_{a}} operators as

*C*(

*x*) commutes with

*C*(

*x*′) for any

*x*,

*x*′, as they are both diagonalized by

*U*. As in the proof of the previous theorem, we know that

*C*(

*x*) is local for

*x*= 0 and we would like to prove locality for every $x\u2208R$. We do this by using the QDB condition to show that also the $x\u0303=1$ is local, and by showing that any other $x\u2208R$,

*C*(

*x*) is a simple function of

*C*(0) and

*C*(1).

*C*is Hermitian, we conclude that

*C*(1) =

*C**, and therefore,

*C** commutes with

*C*=

*C*(0). Finally, a simple algebra shows that

*C*(

*x*) =

*C*

^{1−x}(0) ·

*C*

^{x}(1). Substituting

*x*= 1/2, and using Eq. (34) proves the theorem.

## IV. EXAMPLES

In this section, we describe two exactly solvable models whose dynamics is governed by a Lindbladian satisfying, respectively, the requirements of Theorems 3.1 and 3.2 from Sec. III. In Subsection IV A we describe classical Metropolis-based dynamics that satisfies the conditions of Theorem 3.1, and in Subsection IV B we describe a family of quadratic fermion models that satisfy the conditions of Theorem 3.2. For both models, we prove detailed balance, unique steady-state and constant spectral gap. As the models are exactly solvable, we give explicit expressions for their super Hamiltonians, which are derived independently of the results of Subsection III A.

### A. Model for Theorem 3.1: Classical-like Lindbladians

Here we describe a system that obeys the requirements of Theorem 3.1. First, we specify the steady state, which is the Gibbs state of a classical Hamiltonian, and then write the corresponding Lindbladian that annihilates it; this Lindbladian gives rise to classical thermalization dynamics of the diagonal elements of the density matrix, while dephasing away the off-diagonal elements. The full proofs are given in Appendix B.

*i*as being able to hold a particle (e.g., a fermion or a hard core boson) with some energy

*ϵ*

_{i}, or being empty. The state of the system is then described by a binary string $x\u0332=(x1,\u2026,xn)\u2208{0,1}n$, where

*x*

_{i}determines the occupation of the

*i*th site (either 0 or 1). To achieve a non-trivial Gibbs state, we define a constant interaction

*u*acting between nearest neighbors, which is non-zero if an only if both neighboring sites are simultaneously occupied by a particle. To write down the Hamiltonian of the system, we let $ni=def12(1\u2212Zi)=11i$ denote occupation number operator at site

*i*, and let

*μ*> 0 denote a chemical potential. Our Hamiltonian is then given by

*σ*. It is a geometrically three-local Lindbladian that is given by

*k*∈ {1, …,

*n*},

*b*∈ {0, 1, 2} and {

*γ*

_{k,b}} are some positive (possibly random)

*O*(1) constants. For each

*k*,

*b*, the superoperator $Lk,b$ act locally on qubits

*k*− 1,

*k*,

*k*+ 1 and is defined by:

*L*

_{k,b}are three-local jump operators given by

*k*, and $\Pi k\u22121,k+1b$ is the projector onto the subspace in which the sum of the occupation numbers of sites

*k*− 1 and

*k*+ 1 is equal to

*b*(for

*b*∈ {0, 1, 2}).

We will now show that *σ* is the unique steady state of $L$, and that $L$ is gapped and satisfies QDB. The key is to show explicitly that the representation of $L$ in Eq. (40) is the canonical representation, from which it will follow by Theorem 2.10 that $L$ satisfies QDB with respect to *σ*, and therefore it is a steady state of $L$. Formally, we claim:

*The following properties hold for the Lindbladian* $L$ *defined in (39):*

*The jump operators*${Lk,b,Lk,b\u2020}$*are taken from a modular basis (up to normalization), and so*$L$*is given in the canoniconical**representation.*- $Lk,b$
*satisfy local detailed-balance with respect to**σ**,*$Lk,b\u25e6\Gamma s=\Gamma s\u25e6Lk,b*\u2200s\u2208[0,1].$ *If**γ*_{k,b}> 0*, then*$L$*has a unique steady state**σ**defined in Eq. (36).**If**γ*_{k,b}≥*α*> 0*and*$\u03f5k\u2212\mu ,u=O1$*, then*$L$*is gapped with*$gapL=\Omega \alpha $.

*L*

_{k,b}} is a proper modular basis with well-defined Bohr frequencies given by

*βω*

_{kb}. First, it is easy to see that {

*L*

_{k,b}} are orthogonal and traceless. To see that they have well-defined Bohr frequencies, we need to show that $\Delta \sigma (Lk,b)=e\u2212\omega k,bLk,b$, where Δ

_{σ}(

*L*

_{k,b}) =

*σL*

_{k,b}

*σ*

^{−1}. By Eq. (36),

*L*

_{k,b},

Using Theorem 2.10 it follows that for every *k*, *b*, $Lk,b$ satisfies the QDB condition for *s* ∈ [0, 1/2) ∪ (1/2, 1] (and therefore also for *s* = 1/2) with respect to *σ*, i.e., $\Gamma s\u25e6Lk,b=Lk,b*\u25e6\Gamma s$. As shown in the end of Subsection II C, this implies that *σ* is annihilated by $Lk,b$. Therefore, *σ* is a fixed point of $L$, and, moreover, it is a “frustration-free” Lindbladian.

Showing uniqueness and gap (bullets 3,4) are technically more involved, requiring the use of Knabe and Perron-Frobenious theorems, and are therefore deferred to Appendix B.■

^{19,45,54}As a final remark, we mention that this could be extended to more complicated Lindblad operators, higher dimensional lattices, and other degrees of freedom (e.g., bosons with a larger finite set of allowed occupancies per site or qudits).

### B. Model for Theorem 3.2: Quadratic fermionic Lindbladians

We now describe a family of models that satisfies the requirements of Theorem 3.2. The models consist of free spinless fermions on a lattice, coupled to two particle reservoirs, one with infinitely high chemical potential and one with an infinitely low chemical potential. The high chemical potential reservoir emits particles into the system, and the low chemical potential reservoir takes particles out of the system. The reservoirs are assumed to be large and Markovian, such that they can be integrated out and result in a dissipative Lindblad evolution of the lattice. A physical realization of such settings can be found in, e.g., Sec. 3 of Ref. 55; such Lindbladians can be used for the dissipative generation of topologically non-trivial states.^{56,57}

^{55,58,59}

*a*

_{i}, $ai\u2020$ are, respectively fermionic annihilation and creation operators obeying the standard anticommutation relations, and where

*γ*

^{in},

*γ*

^{out}are positive semi definite matrices with eigenvalues $dkin,dkout$, respectively. Physically,

*γ*

^{in}(

*γ*

^{out}) is responsible for the absorption (emission) of particles from (into) the environment. This model can be seen as a special case of a scenario which is treated by Theorem 3.2: This can be seen, for example, by expanding the fermionic operators in a Majorana basis,

^{58}which is Hermitian, orthonormal and local (in the fermionic sense). Then, expressing the coefficients matrix

*C*in terms of the elements of

*γ*

^{in},

*γ*

^{out}it is easy to show that

*C*·

*C** is gapped if and only if

*γ*

^{in}and

*γ*

^{out}are, and Theorem 3.2 will indicate that the super-Hamiltonian has exponentially decaying interactions. However, we find it more illuminating to not use it, but rather prove directly the properties of this specific model. Under the assumptions pronounced in the claim below, we show that the Lindbldian satisfies QDB with respect to a unique steady state which is Gaussian. This is proven similarly to 4.1, by employing the canonical representation of $L$.

*Let* $L$ *be the Lindbladian in (45). Suppose that* *γ*^{in} *and* *γ*^{out} *are commuting, full-rank matrices.* *Then*

*There is a unique steady state of a Gaussian form:*$\sigma =1Zexp(\u2212Hss),where\u2009Hss=\u2211i,jhijai\u2020aj.$$L$

*satisfies quantum detailed balance with respect to**σ**for any**s*∈ [0, 1].*Let**α*> 0*be a number such that*$\gamma in+\gamma out\u2265\alpha 1$*, then*$gap(L)\u2265\alpha 2$*. In particular,*$L$*is gapped if the minimal eigenvalue of**γ*^{in}+*γ*^{out}*is*$\Omega 1$.

*γ*

^{in}and

*γ*

^{out}, which implies that they can be simultaneously diagonalized:

*γ*

^{in}and

*γ*

^{out}. Notice that in the new basis, $L$ decomposes into a sum of single mode Lindbladians $Lk$, as promised. By calculating the spectrum of each $Lk$, we see that the

*σ*

_{k}. Solving for each $Lk$ it is easy to see that it has the form $\sigma k\u221dexp\u2212\u03f5kck\u2020ck$ with $\u03f5k=def\u2212logdkindkout$. The global steady state is then unique, being the product of all single mode zero states, $\sigma =\u2a02k\sigma k\u221dexp\u2212\u2211k\u03f5kck\u2020ck$. Going back to the original

*a*

_{j}basis completes the proof of bullet 1. Finally, bullet 2 follows from the observation that ${ck,ck\u2020}$ are canonical jump operators, namely, they have well-defined Bohr frequencies:

We remark that: (1) The above can be also proven using correlation matrix formalism and the *continuous Lyaponouv equation*.^{59} (2) Starting from a Gaussian steady state and going to the corresponding single-particle eigenbasis one can show that *γ*^{in}, *γ*^{out} must be diagonal in the same basis. Hence, QDB is satisfied if and only if *γ*^{in}, *γ*^{out} commute.

As a corollary, the super-Hamiltonian can be derived, thus verifying the results of Theorem 3.2 for the current model:

*Let*$L$

*be a Lindbladian as in (45) with commuting*

*γ*

^{in},

*γ*

^{out}

*. Then*$H=\u2212\Gamma \u22121/2\u25e6L\u25e6\Gamma 1/2$

*is given by*

*a*

_{k}yields

*γ*

^{in},

*γ*

^{out}that connect nearest neighbors sites only, i.e., tridiagonal matrices (augmented by the upper-right and lower-left elemetns). Moreover, we consider translation invariant systems (Toeplitz matrices), such that

*c*

_{k}of such a Lindbladian are achieved by diagonalizing

*γ*

^{in}and

*γ*

^{out}, which, due to translation invariance, are given by the a discrete Fourier transform of the original

*a*

_{j}:

*γ*

^{in},

*γ*

^{out}are gapped (in the sense that their smallest non-zero eigenvalue does not vanish as

*n*→ ∞) when $\gamma 0in\u22122\gamma 1in=\Omega 1>0$ and $\gamma 0out\u22122\gamma 1out=\Omega 1>0$, respectively. We also require that $\gamma 0in,\gamma 0out=O1$ for the Lindbladian to have bounded norm. As we now show, the gaps in

*γ*

^{in}and

*γ*

^{out}stated above are responsible for the decay of the matrix elements of $\gamma in\u22c5\gamma out$, and therefore determine the locality of the super-Hamiltonian (47). This is due to the following lemma:

*The matrices in Eq. (48) satisfy*

*where*$d(i,j)=defmin{|i\u2212j|,n\u2212|i\u2212j|}$

*is the metric on the*

*circle.*

As a result, provided that *γ*^{in} and *γ*^{out} are gapped, $H$ given in (47) is geometrically two-local (quadratic) with exponentially decaying interactions. The Proof of Lemma 4.4 is technical and thus left to Appendix C.

We remark that the derivation in Appendix C suggests that when the gap in *γ*^{in} (or *γ*^{out}) closes, that is, when $\gamma 0in=2\gamma 1in$, the super-Hamiltonian becomes long range with polynomially decaying interactions. The degree of the polynomial does not allow an area-law statement for the steady state, according to the results of Ref. 31. See also Fig. 3 for an illustration of the decay in $\gamma in\u22c5\gamma out$.

## V. DISCUSSION AND FURTHER RESEARCH

In this work we have shown how a detailed-balance Lindbladian $L$ can be mapped to a local, self-adjoint superoperator $H$, which we call a super Hamiltonian. The mapping is via a similarity transformation, hence we are guaranteed that the Lindbladian and the super Hamiltonian share the same spectrum (up to an overall minus sign). Moreover, if *σ* is the steady state of the Lindbladian, $\sigma $ is the steady state of $H$. By vectorizing the super Hamiltonian we get a local Hamiltonian whose ground state is $\sigma 1/2$. As a side consequence of our mapping, we also found a necessary condition for a Lindbladian to satisfy detailed balance, which can be checked efficiently.

We observed that local expectation values in *σ* map to local expectation values in the ground state $\sigma 1/2$, and that the mutual information in *σ* is bounded by the entanglement entropy in $\sigma 1/2$. Consequently, several well-known results about the structure of gapped ground states of local Hamiltonians can be imported to the steady state of gapped, detailed-balanced Lindbladians. In particular, we have shown how under mild conditions that can be checked efficiently, the steady state of 1D, gapped, detailed-balanced Lindbladians satisfies an area-law in mutual information, and can be well approximated by an efficient MPO. These results cover many new systems for which the results of Refs. 18 and 21 are not known to apply.

The mapping applies for Lindbladians with traceless jump operators and vanishing Hamiltonian part (a consequence of detailed balance). However, it also applies to Lindbladians with a Hamiltonian that commutes with the steady state, since it leaves the steady state invariant. An example to such a Lindbldian is the Davies generator^{45} that describes a thermalization process. The addition of the corresponding Hamiltonian should not break primitivity,^{60} and we expect that in many cases it will not close the spectral gap.

Our work leaves several open questions and possible directions for future research. First, it would be interesting to see what other results/techniques can be imported from local Hamiltonians to Lindbladians using our mapping. It would also be interesting to see if this mapping can be used in numerical simulations. For example, one can apply DMRG to find the ground state of $H$, and then plug it back to the Lindbladian to see if this is indeed the fixed point. Since $H$ can be easily obtained from $L$, this procedure can be used even if we do not know if the Lindbladian satisfies QDB.

It would also be interesting to further study the various necessary conditions that are needed to prove an area-law for steady states of local Lindbladians, and in particular, find whether the conditions under which our mapping applies can be relaxed. Is there a weaker condition that still ensure a local $H$?

Finally, it would also be interesting to understand if our mapping can be used in the opposite direction. Given a local Hamiltonian, one might ask if it is the vectorization of a super Hamiltonian that comes from some QDB Lindbladian. In such cases it might be possible to probe the ground state of the local Hamiltonian by simulating the time evolution of the Lindbladian on a quantum computer. This might show that the local Hamiltonian problem for this class of Hamiltonians in inside BQP. It would be interesting to characterize this class of Hamiltonians, and see if they can lead to interesting quantum algorithms.

## ACKNOWLEDGMENTS

We are thankful for Curt von Keyserlingk and Jens Eisert for insightful discussions. We are also grateful for an anonymous referee for pointing our attention to the results of Refs. 32 and 33 on the East model. M.G. was supported by the Israel Science Foundation (ISF) and the Directorate for Defense Research and Development (DDR&D) Grant No. 3427/21, and by the US-Israel Binational Science Foundation (BSF) Grant No. 2020072. I.A. acknowledges the support of the Israel Science Foundation (ISF) under the Individual Research Grant No. 1778/17 and joint Israel-Singapore NRF-ISF Research Grant No. 3528/20.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Raz Firanko**: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal). **Moshe Goldstein**: Conceptualization (equal); Methodology (equal); Visualization (equal). **Itai Arad**: Conceptualization (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: PROOF OF LEMMA 3.4

In this appendix, we prove Lemma 3.4. For convenience, we first restate it here.

*Let*

*λ*

_{min}> 0

*be the smallest non-zero eigenvalue of*

*C*·

*C**

*, and assume that*|

*C*

_{ab}| ≤

*J*

*for every*

*a*,

*b*

*. Let*|

*a*−

*b*|

*denote the lattice distance between the supports of*

*P*

_{a},

*P*

_{b}

*. Then*

*where*

*c*

_{1},

*c*

_{2}

*are constants that depend only on the geometry of the lattice and on*

*k*.

Set $A=defC\u22c5C*$, and let *λ*_{max}, *λ*_{min} be the largest and smallest *non-zero* eigenvalues of *A*. Let |*a* − *b*| denote the lattice distance between the support of *P*_{a} and *P*_{b}. Recall that *C*_{ab} ≠ 0 only for *P*_{a}, *P*_{b} that intersect a geometrically local region of *k* sites, and therefore *C*_{ab} ≠ 0 only for |*a* − *b*| ≤ *k*. Similarly, for any integer *ℓ* > 0, $Cab\u2113\u22600$ only when |*a* − *b*| ≤ *kℓ*, and therefore, as also $Cab*\u22600$ only for |*a* − *b*| ≤ *k*, we conclude that $Aab\u2113=(C\u22c5C*)ab\u2113\u22600$ only when |*a* − *b*| ≤ 2*ℓk*.

Following Ref. 61, we assume that there exists a family of polynomial approximations {*P*_{m}(*z*)} to the function $z$ (indexed by their degree) with the following properties:

$|Pm(z)\u2212z|\u2264\tau 1e\u2212\tau 2m$ for every

*z*∈ [*λ*_{min},*λ*_{max}] for some constants*τ*_{1},*τ*_{2}that depend on*λ*_{min},*λ*_{max}, but not on*m*.*P*_{m}(0) = 0.

We will soon find such family, but for now let us discuss its consequences.

*A*is in {0} ∪ [

*λ*

_{min},

*λ*

_{max}], then for any

*m*, $\Vert A\u2212Pm(A)\Vert \u2264\tau 1e\u2212\tau 2m$. Consider now a pair of indices (

*a*,

*b*), and let

*m*be the largest integer for which 2

*km*< |

*a*−

*b*|. Then by the discussion above $Pm(A)ab=0$, and therefore by the triangle inequality,

*m*is the largest integer for which 2

*km*< |

*a*−

*b*|, then 2

*k*(

*m*+ 1) ≥ |

*a*−

*b*| and therefore

*m*≥ |

*a*−

*b*|/(2

*k*) − 1, from which we deduce

*τ*

_{1},

*τ*

_{2}on

*λ*

_{min},

*λ*

_{max}. We will find an

*m*− 1 degree polynomial approximation to $1/z$ in [

*λ*

_{min},

*λ*

_{max}], and then multiply it by

*z*. Following Ref. 10, we use the expansion $1/1+z=\u2211j=0\u221e\u221214j2jjzj$ in the following manner:

*z*∈ [

*λ*

_{min},

*λ*

_{max}], |

*z*/

*λ*

_{max}− 1| < 1, and so the above series converges absolutely. Define

*Q*

_{m}(

*z*) to be the

*m*− 1 degree polynomial that is the sum of the terms in (A2) with degree $\u2264m\u22121$, and let

*R*

_{m}(

*z*) be the sum of all the higher order terms. Then $1z=Qm(z)+Rm(z)$, and using the fact that $2jj\u22644j$, we get

*z*, and setting $Pm(z)=defzQm(z)$, we find that

To conclude the proof, we need to show that *λ*_{max} ≤ *ηJ*^{2}, where *η* is a constant that is a function of the lattice geometry and *k*, independent of the system size. To do that, note that *λ*_{max} = ‖*C* · *C**‖ ≤ ‖*C*‖^{2}. By definition, *C* is a sparse matrix, since at every row *P*_{a} there is only a constant number of *P*_{b} that overlap the same geometrically *k*-local region. Call this constant $\eta $, and note that it only depends on *k* and the geometry of the lattice (i.e., dimension, etc.). Therefore, assuming that |*C*_{ab}| ≤ *J* for all *a*, *b*, we deduce that for any normalized vector *v*, $\Vert Cv\Vert \u2264J\eta $, and so *λ*_{max} = ‖*C*‖^{2} ≤ *ηJ*^{2}.■

### APPENDIX B: CLASSICAL-LIKE LINDBLADIAN: TECHNICAL DETAILS

This appendix is devoted to proving bullets 3 and 4 in Claim 4.1.

#### 1. Proving uniqueness of the steady state (bullet 3)

*x*∈ {0, 1}

^{n}:

*x*by flipping one spin. The corresponding weights are give by:

*b*

_{k}=

*b*

_{k}(

*x*) =

*x*

_{k−1}+

*x*

_{k+1}, and

*k*is defined by the spin that is flipped when

*x*→

*x*′. Note that

*g*

_{x}is responsible for ensuring that $TrL(xx)=0$.

^{62}

*P*

_{xy}, we use the Perron-Frobenious theorem and the connectivity of the Markov chain. Specifically, we show that

*P*

_{xy}(

*t*) > 0 for any

*x*,

*y*,

*t*, and uniqueness will follow as a consequence from Perron-Frobenious (see Theorem 1.1 in Ref. 40).

$Pxy(t)=TrxxeLt(yy)>0$ *for any* *t* > 0.

*G*= (

*V*,

*E*) be a graph where

*V*= {0,1}

^{n}is the set of bit-strings and (

*x*,

*x*′) ∈

*E*if and only if they have Hamming distance

*d*

_{H}(

*x*,

*x*′) = 1 (

*x*′ is obtained from

*x*by flipping one bit). Notice that this is also the connectivity graph of $L$ with self edges omitted. For

*x*,

*y*∈

*V*, define

*ℓ*= (

*ℓ*

_{0}, …,

*ℓ*

_{|ℓ|+1}) to be the shortest path in

*G*from

*x*to

*y*(i.e.,

*d*

_{H}(

*x*,

*y*) = |

*ℓ*|). Take

*c*> 0 such that

*c*>

*g*

_{z}for any

*z*∈

*V*. As a result, $(Lt+c1)xy\u22650$ for any

*x*,

*y*, and in particular $(L+c1)xy>0$ if (

*x*,

*y*) ∈

*E*. Let us expand $eLt$

*d*

_{H}(

*x*,

*y*) ≤

*k*, therefore

*P*

_{xy}receives its first non-zero contributions from the

*k*= |

*ℓ*|th order in the expansion. Moreover, this first contribution is positive, since

*ℓ*and graph connectivity of the semigroup, and the rest of the sum is greater than or equal to zero, hence the whole term is greater than zero. The higher

*k*terms are greater or equal to zero (since all matrix elements are), thus we conclude the claim.■

*f*

_{xy→x′y′}is non-zero only if there are

*k*∈ {1, …,

*n*} and

*b*∈ {0, 1, 2} such that $Lk,bxyLk,b\u2020\u22600$ or $Lk,b\u2020xyLk,b\u22600$ (that is, if

*x*

_{k}=

*y*

_{k},

*x*

_{k−1}+

*x*

_{k+1}=

*y*

_{k−1}+

*y*

_{k+1}), and then it would give

*f*

_{xy→x′y′}=

*f*

_{x→x′}=

*f*

_{y→y′}. The observation is that

*Let* *A*_{ij} *be a matrix with non-negative off diagonal elements and strictly negative diagonal elements, such that* *∑*_{j;j≠i}*A*_{ij} < −*A*_{ii} *for any* *i**. Then* *Spec*(*A*) ⊂ (−∞, 0).

*A*. This is a semi-stochastic matrix (its matrix-elements are non-negative), since

*c*> 0 is chosen such that

*A*

_{ii}+

*c*> 0 for each

*i*. Therefore, $(A+c1)i,j$, and correspondingly $(eA)ij$, are greater or equal to zero for any

*i*,

*j*. Using the Perron-Frobenious theorem,

*e*

^{A}has a maximal eigenvalue

*r*> 0 with an eigenvector

*v*in which

*v*

_{i}≥ 0 for any

*i*. We deduce that

*v*is also an eigenvector of

*A*with largest eigenvalue, due to the monotonicity of the exponent:

*λ*is negative due to the assumption in the claim, and the remaining spectrum will be negative as well (since it must be below

*λ*). Indeed, let

*i*

_{0}= arg max{

*v*

_{i}}, and take

*∑*

_{j≠i}

*A*

_{ij}< −

*A*

_{ii}.■

#### 2. Proving that $L$ is gapped (bullet 4)

To show a gap in the system, we import the *finite-size criteria* for frustration-free Hamiltonians, originally introduced by Knabe^{34} and used, for example, in Refs. 35 and 53. First we introduce the method for generic systems defined on finite dimensional Hilbert spaces, and then use it to show a gap for the system under consideration. We change our notation correspondingly, e.g., we first discuss projectors on a generic Hilbert space and denote them by *P*_{e}, and in the next paragraph we refer to projectors on operators space (super-projectors) which we denote by $Pe$.

*(Finite size criteria)*.*Let*

*G*= (

*V*,

*E*)

*be a regular graph of degree*

*δ*

*. Assign a local Hilbert-space of dimension*

*d*

*to each vertex. Let*

*H*

*be a nearest-neighbors frustration free Hamiltonian defined on the joint Hilbert space of the vertices of the form:*

*where each*

*P*

_{e}

*is a projector defined on the bond*

*e*

*. We define the local gap to be*$\gamma loc=defmine\u2229e\u2032\u2260\u2205gapPe+Pe\u2032$

*. Then*

*H*) ≥

*c*. This is the principle that lies at heart of the method. To achieve a bound of the form (B5), Knabe used the Hamiltonian’s local structure and the local spectral gap, as explained in the following. We start by squaring

*H*and rearranging the double sum:

*H*

^{2}≥

*cH*, we can ignore the rightmost term in (B6), as it is positive semi-definite, being a sum of products of non-overlapping projectors. Using $Pe2=Pe$, one should notice that the first term in Eq. (B6) reduces to

*H*itself. Hence we obtained

*H*

^{2}≥

*H*+

*Q*, where $Q=def\u2211e\u2229e\u2032\u2260\u2205Pe\u22c5Pe\u2032+Pe\u2032\u22c5Pe$. To achieve the desired bound, we write down a lower bound for

*Q*using

*H*. To do so, let us consider

*δ*− 1) factor stems from the fact that for a simple regular graph, each edge intersects 2(

*δ*− 1) distinct edges (

*δ*− 1 on each vertex). On the other hand, due to frustration freeness, we have $Pe+Pe\u20322\u2265gapPe+Pe\u2032Pe+Pe\u2032$, and obtain

*H*

^{2}≥

*c*·

*H*) for $c=2(\delta \u22121)\gamma loc\u22121\u221212(\delta \u22121)$, we see that

*γ*

_{loc}is greater than $1\u221212(\delta \u22121)$, a global gap in

*H*confirmed.■

We remark that the method can be generalized to larger sub-regions and to open boundary conditions.^{34,63}

##### a. Application to $H$

Now we apply the method to the system under consideration to show a global gap in $L$, thus completing bullet 4 of Claim 4.1. First, we reduce the super-Hamiltonian to the sum-of-local-projectors Hamiltonian presented in Eq. (B3) (up to a multiplicative factor). Then, we numerically verify the statement that $\gamma loc>1\u221212(\delta \u22121)=12$ for the 1D ring, which will assure the global gap by Eq. (B7).

*γ*

_{k,b}≥

*α*> 0. Also recall that $H$ is frustration-free, where $Hk,b\u22650$ has a ground state $\sigma $ with energy 0. Then $H$ satisfies

*μ*= 0 and different choices of {

*ϵ*

_{k}} and

*u*(Table I).

Local Gap γ_{loc}
. | |||
---|---|---|---|

Model . | u = −1
. | u = 0.5
. | u = 2
. |

Random {ϵ_{k}} | 0.756 | 0.887 | 0.678 |

Const ϵ_{k} = 1 | 0.769 | 0.913 | 0.778 |

Const ϵ_{k} = 0.5 | 0.755 | 0.891 | 0.698 |

ϵ_{1} = 1, ϵ_{2} = 10 | 0.993 | 0.998 | 0.997 |

Local Gap γ_{loc}
. | |||
---|---|---|---|

Model . | u = −1
. | u = 0.5
. | u = 2
. |

Random {ϵ_{k}} | 0.756 | 0.887 | 0.678 |

Const ϵ_{k} = 1 | 0.769 | 0.913 | 0.778 |

Const ϵ_{k} = 0.5 | 0.755 | 0.891 | 0.698 |

ϵ_{1} = 1, ϵ_{2} = 10 | 0.993 | 0.998 | 0.997 |

We remark that the derivation of the finite size criteria considers two-local interaction (graph interactions), and the model under consideration here is three-local. However, the derivation and results are still valid. This is due to terms that overlap on a single qubit (e.g., $Hk$ and $Hk+2$) commute with each other, as they act trivially on their shared qubits. Therefore, when squaring $H$ in Eq. (B6), their product can be absorbed in the positive “leftover” [the last term in (B6)].

### APPENDIX C: QUADRATIC FERMIONIC LINDBLADIANS: TECHNICAL DETAILS

This appendix is devoted to proving Lemma 4.4, which demonstrates the exponential decay in the coefficients of the super-Hamiltonian with respect to the distance between the real space fermionic operators.

*γ*

^{in},

*γ*

^{out}that connect nearest neighbors (i.e., tridiagonal matrices). For simplicity, we consider translation invariant systems, such that

*a*

_{k}:

*γ*

^{in}(the treatment of

*γ*

^{out}is identical), we notice that

*W*is the (symmetric) random walk matrix $Wij=12\delta j,i\xb11$. Plugging the Taylor expansion $1+x=\u2211k\u22650\alpha kxk$ produces

*i*→

*j*after

*k*steps of the walk, which is less than one and non-zero only when

*k*≥ min{|

*i*−

*j*|,

*n*− |

*i*−

*j*|}; Assume without loss of generality that the minimum is attained at |

*i*−

*j*|. Moreover, the sequence {

*α*

_{k}} is smaller than one in absolute value (polynomially decaying to be precise). Then the matrix entry $(\gamma in)i,j$ satisfies

*γ*

^{in}is gapped and $\Vert \gamma in\Vert =O1$. The analogous argument for

*γ*

^{out}is straightforward. Note that for $\gamma 0in=2\gamma 1in$, the gap in

*γ*

^{in}closes, and a polynomial decay in $\gamma inij$ appears instead (see Fig. 3). This is because for large

*k*’s, the probability for the random walk yields $(Wk)ij=12k+1k|i\u2212j|+k2=Ok\u22121/2$ (see p. 291 in Ref. 64), and also we get that $\alpha k=O1k$. (This can be derived using the expression $\alpha n=(2n\u22123)!22n\u22122n!(n\u22122)!$ and applying the Stirling’s formula for large

*n*.)

*B*

_{1k}for convenience, as it automatically generalized to any

*B*

_{ij}due to translation invariance. Let us write down

*B*

_{1k}, explicitly and use the former bounds for $\gamma in$ and $\gamma out$. Suppose that

*k*≤

*n*/2 as before; then

*m*= 3

*k*/2 + 1, 3

*k*/2 + 2, …,

*n*− 1,

*n*, 1, 2, …,

*k*/2. All we did is to separate the sum to a

*k*/2 length neighborhood of the

*k*th site and the complement (see Fig. 4). At the first sum, the decay is dominated by $2\gamma 1in\gamma 0ind(m,1)=2\gamma 1in\gamma 0inm\u22121\u22642\gamma 1in\gamma 0ink/2$, and at the second sum the $2\gamma 1out\gamma 0outd(k,m)\u22642\gamma 1out\gamma 0outk/2$ term dominates. Therefore we write

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