Local reversibility and entanglement structure of many-body ground states

The low-temperature physics of quantum many-body systems is largely governed by the structure of their ground states. Minimizing the energy of local interactions, ground states often reflect strong properties of locality such as the area law for entanglement entropy and the exponential decay of correlations between spatially separated observables. In this letter we present a novel characterization of locality in quantum states, which we call `local reversibility'. It characterizes the type of operations that are needed to reverse the action of a general disturbance on the state. We prove that unique ground states of gapped local Hamiltonian are locally reversible. This way, we identify new fundamental features of many-body ground states, which cannot be derived from the aforementioned properties. We use local reversibility to distinguish between states enjoying microscopic and macroscopic quantum phenomena. To demonstrate the potential of our approach, we prove specific properties of ground states, which are relevant both to critical and non-critical theories.


I. INTRODUCTION
Gapped ground states define quantum phases of matter at zero temperature. Even though they occupy a tiny fraction of the possible many-body Hilbert space, these states manifest a rich and diverse structure. Standard examples are states with local order parameter such as paramagnetic and ferromagnetic ground states, the superfluid and insulator ones in bosonic and fermionic many-body systems, etc. Other instances, such as the quantum Hall and quantum spin liquids, can arise because of more subtle orders that can be established in the system. A central goal of condensed matter theory is to understand their structure and how it relates to the physics of different phases [1,2]. A natural approach to this problem is to find the constraints that these states satisfy, which set them apart from generic many-body states [3]. To this aim, it is important to understand aspects of locality in these states: to what extent can such states be described by a collection of local degrees of freedom, which are only loosely correlated with each other? On the other direction, analyzing what kind of entanglement these states can harbour, can shed light on the structure of ground states.
An important tool for probing the locality of ground states is the so-called exponential decay of correlations: it is known that gapped ground states on a lattice have a finite correlation length, beyond which the correlations between spatially separated observables decay exponentially [4][5][6][7]. More recently, other quantitative tools have been devised, which characterize the ground state's locality by looking at its entanglement structure [8,9]. A notable example is area law of the entanglement entropy [8], which states that the entanglement entropy of a region with respect to the rest of the lattice should scale like the boundary area of the region rather than its volume. It is expected to hold for all gapped ground states on a lattice, but has only been rigorously proved in one spatial dimension (1D) by Hastings [10] (see Refs. [11][12][13][14] for further results). Hastings' celebrated result yields a complete characterization of 1D gapped ground states as matrix product states (MPS) [15,16], which, to a large extent, provides us a full understanding of the 1D case.
Unfortunately, at higher dimensions our understanding of the problem is still very much limited. Not only that a proof for the area law in higher dimensions is lacking, but it is also unclear how the area law would imply an efficient representation of the ground state [17]. Moreover, when the system has long-range interactions, or it is host in a lattice with a large dimensionality (like an expander graph [18]), the locality properties of the ground state are even more illusive: the exponential decay of correlations no longer holds (since all particles are essentially close to each other), and in general, area law become meaningless as surface areas become as large as volumes.
In this paper we introduce a new notion of locality in many-body quantum states, which we call local reversibility. It is based on the ability to reverse the action of a given perturbation on a quantum state, and we believe that it exposes fundamental features of gapped ground states. We shall see that local reversibility holds for all unique gapped ground states of local Hamiltonians, including systems with long-range interactions or a diverging lattice dimensionality (for which the existing approaches to the locality properties, like the exponential decay of correlation, do not apply). To demonstrate its potential, we study specific problems in many-body physics. We work out rigorous bounds for the quantum fluctuations of locally reversible states. This, in turn, implies new constraints on the critical exponents and rigorous bounds on the quality of the mean-field ansatz, which is often used to treat complicated quantum many- We disturb a quantum state |ψ by an operator ΓL, which is supported in a subsystem L. We then try to recover the state ΓL|ψ by the use of a q-local operator R. If the state |ψ is LR, we can recover the original state by an operator R with q = O( |L|). The entanglement properties of LR states are expected to be highly restricted since entanglement cannot be recovered by local operations once it has been broken. body systems. An important outcome of our approach is an effective way to identify macroscopic quantum entanglement.

II. LOCAL REVERSIBILITY
To motivate our approach, we begin with a heuristic discussion (Fig. 1). Consider a state |ψ that is defined over N localized spins, each with a d-dimensional Hilbert space, and let Γ L be an operator acting on a spin subset L; the total system is given by L ∪ L c with L c the complement of L. Applying Γ L to |ψ , we can potentially disrupt the entanglement between L and L c , even when Γ L |ψ has a constant overlap with |ψ . It is useful to think of |ψ as a superposition of several states |ψ = |ψ 1 + |ψ 2 + · · · and of Γ L as a projector that "kills" some (but not all) of these states. Intuitively, if |ψ contains a global entanglement on the scale of |L| spins, we may only be able to reconstruct |ψ by acting on Γ L |ψ with (at least) an operator that acts non-trivially on the same portion L of the system (i.e., it would be an |L|-local operator). However, when |ψ contains mostly short-range entanglement, we might be able to return to |ψ by using an operator of a much smaller support. How much smaller should that support be for a slightly entangled state? Specifically, as we shall see shortly, the minimal size of support that is needed to reconstruct a product state is O( |L|). This indicates that states that can be reversed by operators of O( |L|) support constitute a class of states with a small amount of entanglement. In the following, we refer to such a class as locally reversible states.
We now put the discussion above on a formal ground.
Definition II.1 (Local Reversibility) We say that a state |ψ is locally reversible (LR) if there exists a function f (x) that decays faster than any power law, such that for every subset of spins L and an operator Γ L defined on it, and for every integer q > 0, there exists a q-local operator such that where · · · is the operator norm.
We note that the q-local operators involved in the definition are operators of the form O := |X|≤q o X , where o X is a local operator supported in a finite subset of spins X = {i 1 , i 2 , . . . , i |X| } with cardinality |X|, which do not necessarily sit next to each other on the lattice. Three remarks are in order. i) Both the shape and the size of L are left completely general. In particular, we can take L to be the entire system (|L| = N ). ii) The presence of the overlap | ψ|Γ L |ψ | in the denominator of the RHS of (1) implies that the local reversibility is essentially a quantum property, non-trivial only at the presence of entanglement. This is because in the classical world, both |ψ and Γ L |ψ are product states in the standard basis (when Γ L is a classical disturbance), and so ψ|Γ L |ψ = 0 unless Γ L does not change |ψ . iii) In some cases, it will make sense to only consider operators R that respect certain symmetries. We will later use this restricted definition of local reversibility for states with symmetry protected topological order (SPTO) [19].
We claim that LR states show high degree of locality, while non-LR states correspond to states with non-local features due to global entanglement. This assertion is formally supported by the following lemma, which shows that quantum fluctuations in an LR state are strongly suppressed. Indeed, consider an LR state |ψ together with a subset of spins L, and let A L be an additive operator of the form A L := i∈L a i . Here, each a i is an Hermitian operator with a i ≤ 1, which acts only on the ith spin. Since the a i operators are commuting with each other, they can be viewed as classical random variables whose joint probability distribution is given by the underlying state |ψ . The following lemma shows that their sum resembles a sum of independent random variables: its probability distribution is strongly concentrated around its mean with a width of O( |L|).
Lemma II.2 Let Π A ≤x and Π A >x be the projectors onto the eigenspaces of A L with eigenvalues ≤ x and > x respectively, and let m be the median of A L with respect to |ψ in the sense that ψ|Π A ≤m |ψ ≥ 1/2 and ψ|Π A ≥m |ψ ≥ 1/2. Then for any positive h, where f (·) is a superpolynomial decaying function defined by the local reversibility of |ψ . An equivalent statement is valid for Π A ≤m−h |ψ . The proof of this lemma is given in Methods section. We note that the width of the distribution is directly related to the appearance of |L| in the definition of local reversibility; an alternative definition with L α in (1), α = 1/2, would lead to a corresponding distribution width of O(L α ) in (2). In that respect, it is insightful to consider the behaviour of Π A ≥m+h |ψ when |ψ is a product state. In such a case, A L can be genuinely regarded as a sum of independent random variables, and consequently, by the central limit theorem, its probability distribution approaches a Gaussian in the limit of |L| → ∞. Crucially, the width of such distribution is O( |L|), which, as mentioned before, proves that if we want product states to be LR, we better take α ≥ 1/2.
An immediate consequence of Lemma II.2 is that the fluctuations of every additive operator A L , which is defined on the entire systems (|L| = N ) must satisfy This inequality has an interesting implication on the 'quantum macroscopicity' of a LR state, as measured by the Fisher information [20][21][22][23]. The Fisher information of a pure state |ψ with respect to an operator A is given by F(ψ, A) = 4 (∆A) 2 [22]. In Ref. On the other hand, the converse is not true: there are states with N eff = O(1) that are also non-LR. For instance, as we shall see, degenerate, topologicallyordered states, turn out to be non-LR but still satisfy inequality (2), namely N eff = O(1).

III. REVERSIBILITY OF GROUND STATES
We now introduce our main tool for identifying LR states. The following theorem states that unique gapped ground states of local Hamiltonians are LR. It holds for a very wide class of quantum systems that are described by k-local Hamiltonians, therefore with interactions that may involve k spins (note that k is not necessarily equal to q from the definition of the operator R above) of the form where g is a constant of O(1). We implicitly assume that the spins sit on a lattice, but we make no direct use of the lattice structure or its dimensionality. Instead, we use the second condition in (4), meaning that the total strength of all interactions in which the ith spin participates is bounded by a constant of O(1). This definition of H captures a very wide class of quantum systems; in particular, it includes not only short-range interacting systems but also long-range interacting systems. We denote the ground state of H by |Ω , and fix its energy to be E 0 = 0. The rest of the energies are denoted by Finally, we let δE := E 1 − E 0 be the spectral gap just above the ground state. With this notation at hand, our main theorem is given as follows.
Theorem III.1 For every spin subset L and every operator Γ L defined on it, and for any positive integer q, there exists a q-local operator R that satisfies where n 0 := q/k and Inequality (5), together with the definitions of n 0 and ξ, and therefore |Ω is LR when δE = O(1). Hence the existence of a spectral gap places strong restrictions on structure of the ground states for very wide class of Hamiltonians.
Here we summarize the idea of the proof and defer the details to Methods section. Using recent results from Ref. [24], we conclude that after applying the operator Γ L to the ground state |Ω , we get a state which consists mainly of excitations with energies of at most O(|L|). Beyond that scale, the weight of the excitations decays exponentially. This is shown schematically by the blue curve in Fig. 2. Then following ideas from a recent new proof of the 1D area law [12], we construct the operator R by approximating the ground-state projector using a polynomial of H. This polynomial is essentially a scaled version of the Chebyshev polynomial (red curve in Fig. 2), chosen such that it approximately behaves as a boxcar function in the range [δE, 2E c + δE], thereby suppressing the majority of excitations in Γ L |Ω . Crucially, even though it rapidly increases for x ≥ 2E c +δE, this blowup is cancelled by the exponential decay of the high-energy excitation.

IV. EXAMPLES OF LOCALLY VS. NON-LOCALLY REVERSIBLE STATES
Let us now apply the theorem above to several exemplary states emerging in different contexts. The list of states is summarized in Table I. In particular, we will demonstrate how local reversibility implies the absence of macroscopic entanglement. We begin with LR states.
. As H is made of commuting projectors, its spectral gap is necessarily δE = 1. 2. Graph states with bounded degree. These states are defined on a graph in which each node has at most O(1) neighboring nodes [25,26] . The graph state is a non-degenerate gapped ground states of a Hamiltonian which is the summation of the following commuting stabilizers [27] are the Pauli matrices and {j 1 , j 2 , . . . , j ki } are nodes which connect to the node i. By assumption, k i = O(1), and hence the Hamiltonian is O(1)-local. By the commutativity of its terms, we conclude that it has a spectral gap δE = O(1), and so by Theorem III.1 such graph states are LR. 3. Short-range entanglement (SRE) states. The third example are states that can be obtained by a constant-depth quantum circuit acting on a product state. In the literature they are often dubbed as "trivial states" [28,29], or "short-range-entanglement (SRE) states" [2]. A constant-depth quantum circuit is a unitary operator that can be written as a product of k = O(1) unitary operators U = U 1 · · · U k where each unitary U i is given as a product of unitary operators To see why these are LR states, we write |ψ = U |φ , where U is the constant-depth circuit, and |φ = |φ 1 ⊗|φ 2 ⊗· · · is a product state. Then it is easy to see that for any operator O with a support of O(1), U OU −1 has also an O(1) support, and therefore if H is a local Hamiltonian for which |φ is the unique ground state (see the first example), then H = U HU −1 is also a local Hamiltonian. Furthermore, H has the same spectrum as H, and so it is gapped with the unique ground state, which is exactly |ψ . By Theorem III.1 this state is LR.
We note that not all LR states are also SRE states, or, equivalently, long-range entanglement (LRE) does not necessarily imply non-LR. For example, Kitaev's toric code [30] on a sphere is a commuting local Hamiltonian and has a non-degenerate ground state with an O(1) gap, and therefore by Theorem III.1 it is LR. Nevertheless, it cannot be generated by a constant depth circuit working on a product state, and is therefore not an SRE state [31].
We now turn to non-LR states. So far, we identified two main classes of such states: states with the large Fisher information and topologically ordered states. Both of them show global non-locality; the former ones can be characterized by the scaling of local observables such as the magnetization, while the latter ones cannot.
5. States with Fisher information of O(N p ) with p > 1. This result comes directly from Lemma II.2. A quintessential example of this class is the GHZ state [22], which has the scaling with p = 2.
6. States with degenerate topological order. The Fisher information cannot detect a locally hidden order such as the topological order, where we can have p ≤ 1 for any local observables. Even in this case, we can still see its non-LR by taking the subsystem L large enough. We demonstrate this point in the methods section by considering Kitaev's toric code model on a torus [30]. Essentially, the idea is that by taking L to be a nontrivial loop in the torus, the perturbation Γ L can take a ground state to another ground state. Yet, due to the topological nature of the system, this action cannot be reversed by any operator R that is a sum of O( |L|)-local operators, since all of them are necessarily supported only on trivial loops.

7.
States with a degenerate symmetry protected topological order. The same arguments showing that degenerate topologically ordered states are not LR can be applied to the case of degenerate symmetry protected topological order. Such states show topological order only to a restricted set of operators defining a certain symmetry G [19]. They cannot be adiabatically connected to a product state using only operators from G, and in that restricted sense they are not SRE (see the Methods section for the definition). An important example of states with SPTO can be obtained from graph state's Hamiltonian on an open lattice, where one removes the boundary stabilizers. This removal introduces degeneracy to the groundspace. Much like the case of Kitaev's toric code, we can also show here that the resulting ground states are non-LR as long as we restrict the operator R to satisfy the symmetry of the graph Hamiltonian without the boundary stabilizers. We refer to these states as symmetry restricted non-LR states. We present an example of such states for 1D case [32] in the Methods section. Theorem III.1, together with Lemma II.2 provides a remarkable insight into the structure of unique ground states; for any such ground state |Ω , and for any additive operator A L = i∈L a i defined on a spin subset L, with c 1 a constant of O(1). It is interesting to contrast this inequality with the corresponding statistics of a product state. In such a case, A L can be viewed as a sum of independent random variables, and by the Chernoff bound [33], Π A ≥m+h |ψ ≤ e −O(h 2 /|L|) . In this sense, unique gapped ground states enjoy a weaker, yet still non-trivial, notion of local independence.
It is also worth noting that this independence cannot be (at least directly) deduced from the exponential decay of correlation of gapped ground states [4, 5, 7, 34], since it can be applied to sets of observables that may sit very close to each other on the lattice. Moreover, we can apply it to systems with long-range interactions, such as the Lipkin-Meshcov-Glick model [35] and systems defined on the expander graphs [18], in which the maximal distance between any two spins is O(1) and O(log N ), respectively. Finally, we remark that inequality (7) can be extended to a more general setting; we can derive a similar bound for low-lying energy states, i.e., not necessarily the exact ground state (T.K., I.A., L.A. and V. V., manuscript in preparation).
A simple consequence of inequality (7) is a trade-off relationship between the spectral gap and the fluctuation ∆A L := ( Ω|A 2 L |Ω − Ω|A L |Ω 2 ) 1/2 of A in the ground state: This has two interesting implications: 1. Bounds on the critical exponents. Let us consider the critical regime, δE → 0. Define A L = N i=1 a i with L a total system and {a i } N i=1 order parameters (e.g. magnetization). We then introduce the critical exponents z, η, γ and ν as in Refs. [36]; z is the dynamical critical exponent, η is the anomalous critical exponent, γ is the susceptibility critical exponent and ν is the correlation length exponent. By applying the finite-scaling ansatz [36,37] to (8), we can obtain where the second equality comes from the Fisher equality 2 − η = γ/ν. We remark that (9) holds for very general settings both for homogeneous and disordered critical systems (see [38] for a non trivial example where our inequality can be applied). Incidentally, we note that (8) gives non-trivial bounds for the critical Lipkin-Meshcov-Glick model, a system with long-range interactions [35]. The details of this calculation are given in Appendix B 4. 2. Validity of mean-field approximations. The second implication of inequality (8) concerns the validity of the mean-field approximation. Just as the first implication, the full details are given in the Appendix B 5. The idea is that since the operators A L in (8) are arbitrary (as long as they are additive on L), we can use them to probe the two-spin reduced density matrix ρ ij and its relation with its mean-field approximation ρ i ⊗ ρ j . Specifically, it can be shown that for every spin subset L and an arbitrary spin i outside of it, This implies that on average, for each spin j ∈ L, ρ ij −ρ i ⊗ρ j ≤ O(1/ |L|δE). If our system is defined by a nearest-neighbor two-body Hamiltonian on a regular grid with coordination number Z (the number of neighbors of each spin), then taking L to be the set of neighbors (|L| = Z), one immediately obtains a bound on the quality of the mean-field approximation for the energy density for ∀i: where the sum is taken over the spins adjacent to i.
We therefore obtain a quantitative bound on how the error of the mean-field approximation decreases as the lattice dimension (on which the coordination number depends) goes to infinity. This result is consistent with the folklore knowledge in condensed-matter physics that the mean-field becomes exact in infinite dimension. Recently, similar results have been obtained in different manners by Brandão et al. [39] and Osterloh et al. [40] VI

. SUMMARY AND OPEN QUESTIONS
In this work, we introduced a new notion of locality in quantum states, the local reversibility, which is defined in terms of the type of local operations that are needed to reverse the action of perturbations to the state.
We proved that all unique ground states of gapped local Hamiltonians are locally reversible (Theorem III.1), and, on the other hand, we showed how local reversibility implies a suppression of quantum fluctuations (Lemma II.2). Together, these two results provide new insights into the structure of unique ground states of gapped local-Hamiltonians: i) a low Fisher information, which is an indication for the lack of quantum macroscopicity in these states; ii) a novel inequality for the critical exponents in these systems; iii) a quantitative analysis of the mean-field approximation; and finally, iv) since an adiabatic (local unitary) evolution of product states is locally reversible, our result clearly implies that all the gapped quantum phases of matter, disordered or with local order parameter (Landau symmetry breaking quantum phases), are reversible. In contrast, degenerate topological phases or the symmetry protected topological phases, are not reversible.
Our work leaves many open questions for further research. First, it would be interesting to know whether Theorem III.1 is tight, or whether the exponential decay in the RHS of inequality (5) can be replaced with, say, a Gaussian. If such stronger bound can be proved, it would show via Lemma II.2 that the statistics of additive operators on a LR state is the same as that of a product state. A recent proof of the Berry-Esseen theorem for the quantum case by Brandão et al. [41] hints that this might be the case.
Another interesting direction to explore is whether local reversibility, or one of its consequences, such as Lemma II.2 or inequality (8), can be incorporatedexplicitly or implicitly -in the construction of tensor networks in higher dimension (e.g., Projected entangled pair state, or PEPS [16]). By construction, these states satisfy the area-law, but we now know that they should also satisfy local reversibility. This may speed up the contraction of such tensor networks, which is the main bottleneck in the variational algorithms [42,43]. It would be also interesting to see if local reversibility could be used to actually prove that PEPS are indeed faithful representations of gapped ground states. A good place to start studying this question is in the 1D world. We know that MPS can describe both LR and non-LR states (i.e., GHZ). The natural problem is then to pinpoint what is needed for an MPS to describe an LR state.
Related to that, local reversibility might be useful for proving the area-law conjecture in 2D and beyond. In particular, in Ref. [14], Brandão et al. have proved the 1D area law only from the exponential decay of bipartite correlations. It might be possible to improve Brandão et al. result by using local reversibility, and this may pave the way for new insights regarding the area-law conjecture in higher dimensions.
Finally, it would be interesting to understand if local reversibility could somehow be used to characterize unique gapped ground states. In other words, is local reversibility also a sufficient condition for unique gapped ground states? Strictly speaking, this is incorrect, as there are LR states which are not gapped ground states. For example, the state |000 · · · 0 + (N )|111 · · · 1 where (N ) decays faster than any polynomial is trivially LR, but can never be a unique gapped ground state of k-local Hamiltonians as long as k ≤ N/2 (see Ref. [44]). Nevertheless, we may still ask if, in some sense, every LR state can be approximated by a unique gapped ground state. If this is not the case, it would be interesting to understand which are these LR states that cannot be even approximated by gapped ground states. ACKNOWLEDGEMENT We are grateful to Naomichi Hatano, Tohru Koma, Hal Tasaki, Taku Matsui, Tomoyuki Morimae and Dorit Aharonov for helpful discussions and comments on related topics. We also thank Naomichi Hatano for valuable comments on the manuscript. This work was partially supported by the Program for Leading Graduate Schools, MEXT, Japan. TK also acknowledges the support from JSPS grant no. 2611111. Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant random numbers from quantum processes.  Set Γ L := Π A ≤m and use the local reversibility of |ψ with q := h/2 − 1. As Π A ≤m = 1 and ψ|Π A ≤m |ψ ≥ 1/2, we see that there exists a q-local operator R such that RΠ A ≤m |ψ − |ψ ≤ 2f (q/ |L|). Using this, together with the fact that Π A ≤m = 1 and the triangle inequality, we get Π A ≥m+h |ψ ≤ Π A ≥m+h RΠ A ≤m |ψ + 2f (q/ |L|). To finish the proof we will show that Π A ≥m+h RΠ A ≤m = 0. This follows from the fact that A L is a sum of (commuting) 1-local operators of norm 1, and therefore every q-local operator can take an eigenvector |a of A L with eigenvalue a to a superposition of eigenvectors c a |a with |a − a| ≤ 2q. But as h > 2q = 2( h/2 − 1), we conclude that the overlap Π A ≥m+h RΠ A ≤m necessarily vanishes. This proves the lemma.

Proof of Theorem III.1 (sketch)
The proof of Theorem III.1 is rather technical, and therefore we only sketch it here, giving the full details in Appendix B 1.
Multiplying inequality (5) by | Ω|Γ L |Ω |, and writing for brevityR := Ω|Γ L |Ω R, we obtain So for the state to be LR, we need to find aR whose action on Γ L |Ω approximates the action of the ground state projector |Ω Ω| on it. In addition, in order to satisfy the premise of the theorem, it has to be a qlocal operator. To this aim, we look for a low-degree polynomial F R (x) and writeR := F R (H). Specifically, choosing a polynomial of degree n 0 := q/k guarantees that it will contain at most q-local terms, since, by definition, each term in H is k-local.
To understand the restrictions on F R (x) that inequality (A1) poses, it is convenient to work in the energy basis {|E }: expanding Γ L |Ω = E c(E)|E , we want i) F R (0) = 1 (recall that have set E 0 = 0), and ii) E≥δE |c(E) · F R (E)| 2 1/2 ≤ 10 Γ L e −2n0/ξ . This is achieved using two ideas, which are demonstrated in Fig. 2.
The first idea is that the expansion of Γ L |Ω is dominated by energies of at most O(|L|); beyond that scale, c(E) is exponentially decaying. This is a direct corollary of Theorem 4.1 in Ref. [24], which for our case implies: Corollary A.1 (from Theorem 4.1 in Ref. [24]) Let Π H ≥E be the projector into the eigenspace of H with energies greater than or equal to E. Then In Ref. [24], this theorem was proved under the more restricted condition that every particle participates in at most g interactions of norm 1, but this can be easily relaxed to the current condition, given in definition (4). The bound in (A2) implies that our polynomial should mainly "kill" the energy excitations of Γ L |Ω in the range [δE, O(|L|)]. Following Ref. [12], we let F R (x) be the n 0 th order Chebyshev polynomial [45], scaled such that x : [−1, 1] → [δE, 2E c + δE] and F R (0) = 1. As proved in Appendix B 1, this polynomial fluctuates between ±e −2n0/ξ in the range [δE, 2E c +δE], and then diverges like O((2x/E c ) n0 ). It is our choice of E c in Theorem III.1 which guarantees that this divergence is cancelled by the exponential decay of Corollary A.1. After a rather straightforward calculation, one can show that total contributions of the energy segments [δE, 2E c + δE] and [2E c + δE, ∞) to (R − |Ω Ω|) · Γ L |Ω is exponentially small. In Kitaev's toric code model [30] there are four degenerate ground states {|Ω , T L1 |Ω , T L2 |Ω , T L1 T L2 |Ω } which are characterized by topologically non-trivial loop operators T L1 and T L2 . Here, L 1 , L 2 are the two nontrivial loops on the Torus, where |L 1 | = O( ), |L 2 | = O( ) with the system length, namely = O( √ N ). We show that the ground state |Ω is not LR. It is known that the state |Ω is robust to any disturbance Γ L as long as |L| . However, if we take a subregion of size |L| as large as O( ), the state is no longer robust as we will show below.
Let us now define |ψ ± := 1 √ 2 (|Ω ± T L1 |Ω ). Then a central property of topological order [46] is that there exists an O(1) constant c such that for any operator o X with support |X| ≤ c , Consequently, we also have ψ + |o X |ψ − = 0, and therefore, for any q-local operator R with q ≤ c we have ψ + |R|ψ − = 0. Now, take the operator Γ L := ( which means that Γ L |Ω cannot be reversed to |Ω as long as q ≤ c = O(|L|), implying that |Ω is non-LR.

Symmetry-restricted Local Reversibility.
Symmetry restricted LR states (SRL) can be introduced along very similar lines used in Sect.II. Let's consider a given Hamiltonian H enjoying a global symmetry G; let |ψ be the ground state of H. We say that the state |ψ is SLR iff the property (1) holds with a q-local operator R enjoying the same symmetry group of the Hamiltonian: [R, G] = 0.
Here we present an example of states which are not SLR. Cluster states provide an example of SPTO. The 1d cluster states [47] are the ground states of the Hamiltonian which enjoys a global symmetry Z 2 × Z 2 [32]. With the boundary conditions σ x 0 = σ x L+1 = I, the ground space of H C is unique with a spectral gap. For σ x 0 = σ x L+1 = 0, in contrast, the ground space is four-fold degenerate because the two stabilizers (out of L) σ x 0 σ z 1 σ x 2 and σ x L−1 σ z L σ x L+1 can be fixed at will [32]. Let {|Ω α , α = 0, 1, 2, 3} be spanning the ground state manifold. Due to the symmetry-protected topological order of the system, it follows that the ground states |Ω α cannot be distinguished by any local operator o X in Z 2 × Z 2 : Ω α |o X |Ω α = Ω β |o X |Ω β , and Ω α |o X |Ω β = 0. (A4) with |X| ≤ cN (c = O(1)). Using these conditions, the symmetry-restricted non-LR of |Ω α follows from the same arguments that were used in the proof of the non-LR of the toric code. Following the proof's sketch in the methods section, we start from inequality (A1) in the main text. Our goal is to find a polynomial F R (x) such that the action of the operatorR := F R (H) on the state Γ L |Ω approximates the action of the ground state projector |Ω Ω| on it. As H is a k-local operator, choosing n 0 := q/k guarantees thatR is a q-local operator.
Working in the eigenbasis of H, we expand Γ L |Ω = E c(E)|E , and as F R (H) is diagonal in this basis, Therefore, for inequality (A1) to hold, it is sufficient that As noted in the outline of the proof in the methods section, to prove these properties we use two ideas. The first is that the weight of the high energy excitations in Γ L |Ω decays exponentially, as shown in Corollary A.1. The second is to take F R (x) to be a scaled version of the n 0 'th order Chebyshev polynomial. Let us start from the second idea. The nth order Chebyshev polynomial [45] of the first kind is given by Equivalently, for x ∈ [−1, 1] it is given by T n (x) = cos(n arccos(x)), and for |x| > 1 by T n (x) = cosh(n arccosh(x)). What makes the Chebyshev polynomial so useful to our purpose are the properties that are summarized in the following lemma, whose proof is given in Sec. B 2 Setting ξ := 1 + 2E c δE , and E c := 3g|L| + 16gkn 0 , (B7) we define F R (x) to be the polynomial .

(B8)
In other words, we defined it to be the n 0 th order Chebyshev polynomial, scaled such that and for x ≥ 2E c + δE, For brevity, we define the low and high energy ranges I LOW := [δE, 2E c + δE) and I HI := [2E c + δE, ∞). Then using the triangle inequality, we split the sum in the LHS of (B2) , and bound each term separately. The low-energy term is bounded by which follows from Inequality (B9) and the fact that To finish the proof, we will show that the high energies term is upper bounded by 8 Γ L e −2n0/ξ . To this aim, we write I HI = I 1 ∪ I 2 ∪ I 3 ∪ . . ., where I j := [2E c + δE + (j − 1)η, 2E c + δE + jη) and η is a positive constant which will be set afterward. Using the triangle inequality once more, we get As |F R (x)| monotonically increases for x ≥ 2E c + δE (which follows from the fact that the Chebyshev polynomial is monotonic for x ≥ 1), it follows that To bound the other term, we use Corollary A.1 from the main text, which gives us where we have defined Together, this gives us The final step is to show that for x ≥ 2E c + δE, (see Subsection B 3 for a proof), which leads to Summing over all j ≥ 1, then gives us Using the definition of E c in Eq. (B7), we find that e −λ(2Ec+δE−6g|L|)/2 = e −λ(16gkn0+δE)/2 ≤ 1, and calculating the geometrical sum we get e λη ∞ j=1 e −jηλ/2 = e λη/2 /(1 − e −λη/2 ), which can be minimized to 4 by choosing η such that e λη/2 = 2. All together, we therefore get which completes the proof.
To prove inequality (B5), consider the general inequality which is valid for any x ≥ 1 and 0 ≤ y ≤ 1 (the inequality can be proved by differentiating (2x) n − (2x − y) n + y n with respect to x, and noting for x ≥ 1 and 0 ≤ y ≤ 1 it is a monotonically increasing function of x, and its minimum value 0, which is obtained for x = 1 and y = 0). Choosing y = x − √ x 2 − 1, the LHS of inequality (B15) becomes 2T n (x), which proves (B5).

Critical exponents
Here, we derive inequality (9) for the critical exponents z, η, γ and ν under the scaling ansatz (B17) [36,48]. Recall that we are considering a local Hamiltonian system at T = 0 which is driven towards critically, and let A = i a i , where a i are single particle operators that correspond to a local order parameter (e.g., spin localized at site i leading to the magnetization along a given axes). Our starting point is inequality (8), namely δE · (∆A) 2 ≤ const × N . (B16) We first define the variance (∆A t ) 2 which depends on time as (∆A t ) 2 := A(t) − A · A − A , where A(t) = e −iHt Ae iHt . The variance (∆A t ) 2 reduces to the summation of the correlation functions: where a i (t) := e −iHt a i e iHt for i = 1, 2, . . . N . Note that (∆A t=0 ) 2 is equal to (∆A) 2 = A 2 − A 2 . In the following, we denote C i,j (t) = C(r, t) under the assumption of the translation symmetry. Now, we adopt the following scaling ansatz [36]: S(q, ω; ξ) = ξ 2−η D(qξ, ωξ z ), where ξ is the correlation length and S(q, ω; ξ) is the spatial-temporal Fourier component of C(r, t), namely S(q, ω; ξ) = r t C(r, t)e −i(q·r+ωt) drdt. (B18) We also define S(q; ξ) as S(q; ξ) = 1 2π ∞ −∞ S(q, ω; ξ)dω.
We then obtain the scaling of S(q = 0; ξ) ∝ ξ 2−η−z by taking the scaling (B17) for S(q, ω; ξ), and hence we have (∆A t=0 ) 2 /N ∝ ξ 2−η−z . We also have the scaling of the energy gap as δE 0 ∝ ξ −z [36] by the use of the dynamical critical exponent z. At a critical point, where the correlation length is as large as the system length, the inequality (B16) reduces to −z ≤ −(2 − η − z) (B20) in the infinite volume limit (N → ∞). This reduces to the inequality (8) in the main manuscript. We close the section applying inequality (8) to a system with long-range interactions: the Lipkin-Meshcov-Glick model H LMG = − λ N i<j (σ x i σ x j + γσ y i σ y j ) + N i=1 hσ x i with |γ| ≤ 1. At the critical point λ = |h|, we have the scaling [49] of δE ∝ N −1/3 and (∆M x ) 2 ∝ N 4/3 , where M x is the magnetization in the x direction, M x = N i=1 σ x i . Thus, the spectral gap and the fluctuation can give the non-trivial sharp upper bounds to each other.