Local constants of motion imply information propagation

Interacting quantum many-body systems are usually expected to thermalise, in the sense that the evolution of local expectation values approach a stationary value resembling a thermal ensemble. This intuition is notably contradicted in systems exhibiting many-body localisation, a phenomenon receiving significant recent attention. One of its most intriguing features is that, in stark contrast to the non-interacting case, entanglement of states grows without limit over time, albeit slowly. In this work, we establish a novel link between quantum information theory and notions of condensed matter, capturing the phenomenon in the Heisenberg picture. We show that the existence of local constants of motion, often taken as the defining property of many-body localisation, together with a generic spectrum, is sufficient to rigorously prove information propagation: These systems can be used to send a signal over arbitrary distances, in that the impact of a local perturbation can be detected arbitrarily far away. We perform a detailed perturbation analysis of quasi-local constants of motion and also show that they indeed can be used to construct efficient spectral tensor networks, as recently suggested. Our results provide a detailed and model-independent picture of information propagation in many-body localised systems.

When driven out of equilibrium, generic interacting manybody systems are usually expected to thermalise [1][2][3], in the sense that local expectation values can be described by thermal ensembles. For this to be at all possible, local expectation values need to equilibrate to an apparent stationary state and energy has to be transported through the whole system. Such an expected generic behaviour is prominently violated by many-body localised systems [4] that show a strong suppression of transport [5][6][7] and fail to serve as their own heat bath [8,9]. Thus, these systems do not thermalise and energy remains largely confined within certain regions.
On the level of static properties of the Hamiltonian, equilibration in expectation is guaranteed by non-degenerate energy values and gaps [10][11][12]; a condition that is expected to hold as soon as small random interactions are added to the system. While these equilibration results are by now rather well understood, the question to what extend the local equilibrium values can be captured by thermal ensembles is still open to debate. A direct way to ensure thermal behaviour is given by the eigenstate thermalisation hypothesis [13][14][15], one reading of which assumes that most individual eigenstates are already highly entangled and are locally indistinguishable from Gibbs states.
For many-body localised systems, the static properties are markedly different. While the randomness typically occurring in these models will almost surely guarantee non-degenerate energy values and gaps, the individual eigenstates generically have low entanglement [8,9,16] and are expected to be efficiently described in terms of tensor networks [16][17][18]. Moreover, one typically finds that the system has local constants of motion [19][20][21] that are invariant in time. In fact, it has been shown that such local constants of motion can be used to infer the structure of the eigenstates and obtain an efficient tensor network description of the eigenprojectors [18].
The investigation of transport properties in interacting many-body systems has a long tradition, with upper bounds, giving an effective speed of sound, being provided early on by Lieb and Robinson [22]. For localising systems, the noninteracting case notably leads to a full suppression of trans-port, at least in the limit of infinite systems. It came as some surprise that this is no longer the case in the presence of interactions and that entanglement entropies grow without limit over time [6]. These numerical findings indicate that transport is allowed in these models, at least for the infinite energy states usually considered.
In this work, we present a rigorous proof for information transport in many-body localised systems, using only the existence of local constants of motions and a generic spectrum. Our approach is fully model independent and assumes no specific structure of the Hamiltonian. This is achieved by basing the proof on a recently established link between transport and equilibration behaviour [3,23]. Our results are a considerable step forward in the general quest for lower bounds for transport, which so far have only been obtained in highly specific systems.

Many-body localisation
In this work, we will focus on systems exhibiting manybody localisation (MBL). While a comprehensive definition of this phenomenon is still lacking, it is generally expected that it is closely connected to the existence of approximately local constants of motion [8,18]. These are operators that commute with the Hamiltonian [H, Z] = 0, but are nevertheless to some extent local. Locality of these objects is best captured in terms of a reduction map that restricts the corresponding operator to a certain region S (Appendix 1 for details). For local operators, we assume a central support region X, enlarged regions X l (Fig. 1)  the enlarged region X l . We choose the following description where g is some rapidly decaying function and · denotes the operator norm. Naturally for strictly local observables, the initial region X needs to be taken large enough to include the full support.
In order to further investigate the structure of the local constants of motion, we now turn to two simple models of manybody localisation. In one setting, it is assumed that the Hamiltonian is simply a sum of these commuting approximately local terms [7] H = jh j . (1) Another possibility that has been extensively studied is a Hamiltonian that is a higher order polynomial in terms of approximately local constants of motion [20,21] where J i,j ∈ R for all i, j and the interactions decay both with their order and with the distance of the involved spins. There is a very important difference between these two models. Due to the randomness typically present in many-body localising systems, it is strongly expected that the spectrum of the corresponding Hamiltonians is generic and in particular has non-degenerate energies and gaps. To achieve this in the former Hamiltonian in Eq. (), each local constant of motion already has to have a complicated spectrum. In contrast, the approximately local constants of motion in the second Hamiltonian (Eq. ( 2)) are expected to behave similar to simple Pauli-Z-matrices and have a small number of distinct eigenvalues. Due to the higher order interactions in Eq.
( 2), this is still perfectly compatible with a generic spectrum of the full Hamiltonian. We will follow the intuition provided by the Hamiltonian with higher order interaction between the constants of motion, which leads us to the following definition.
Definition 1 (Local constant of motion). Let Z be an operator that commutes with the Hamiltonian and has M disjoint eigenvalues, all separated by a spectral gap lower bounded by γ > 0, independent of the system size. Z is an exactly local constant of motion, iff it is strictly local and an approximately local constant of motion, iff it is approximately local.
Following Ref.
[7], we will refer to the approximately local objects as qIOM (quasi-local integral of motion) and the exactly local objects as sIOM (strictly local integral of motion). One of the first features of the simple spectrum of the constants of motion is that we can deduce that the dimension of the eigenspaces has to grow exponentially in the system size. Moreover, using such constants of motion also allows to prove that the spectral tensor networks constructed using exactly local constants of motion [18] can still be carried out in the case of where they are only approximately local (Appendix 4).
This leads to the interesting setting that for systems exhibiting MBL, eigenstates will typically efficiently be captured in terms of matrix-product states with low bond dimension. Yet, product states will build up arbitrary large entanglement over time [6]. We now turn to our main result, namely a rigorous proof of information transport in MBL systems.

Main result: Proof of transport
In order to capture transport in these models, we will start with a strictly local observable A and evolve it in the Heisenberg picture. The support of an observable describes the area, where it can potentially detect local excitations. Hence, transport can then be captured by how well the time evolved operator can be truncated to certain regions.
Definition 2 (Transport on average). We will say that a Hamiltonian has transport on average, if we can fix an error arbitrarily close to unity and there still exists a strictly local observable A with unit operator norm, such that, for each finite region S, truncation of the Heisenberg evolution to that finite region necessarily leads to this fixed error, as long as the system is large enough. More precisely, we have that Here · denotes the infinite time average. This definition is very restrictive, in the sense that it demands that the error can be chosen arbitrarily close to 1. On the other hand, it does not require any information for the corresponding time, but rather only refers to the time average, which is connected to the overwhelming majority of times. In particular, it is well possible for the support of the observable to take exponentially long to grow.
The above definition is a natural way to capture information transport, as it implies that we can find a state vector |ψ , such that a local excitation created by some local unitary V will be, on average, detectable at distances arbitrary far away Schematic sketch of our main result. The existence of a local constant of motion Z, drawn in blue, together with a suitably nondegenerate spectrum implies transport, in the sense that there exist local observables whose support spreads without bounds in the Heisenberg picture.
This follows directly from rewriting the reduction map in terms of a unitary twirl over the Haar measure (Appendix 1). Our main result states that this information transport can be rigorously deduced from only the existence of a local constants of motion and a suitably non-degenerate spectrum of the Hamiltonian.
Theorem 1 (qIOM imply transport). Let H be a Hamiltonian with non-degenerate energies and gaps and Z be a qIOM with decay function g with localisation region X, spectral gap γ > 0 and eigenspaces with dimension larger than d min . Then H necessarily has transport on average in the sense that there exists a local operator A initially supported on X l ⊃ X with A = 1 such that A t , on average, has support outside any finite region S We remark, that the first non-constant term in Eq. (1) can be chosen arbitrarily small by picking the initial support X l large enough and the second term decays exponentially with system size L, due to the growth of the degeneracy d min .
This result shows that for many-body localising Hamiltonians, a zero velocity Lieb-Robinson bound does not occur and it is always possible to use the system to transport information. The results does not rely on any specific form of the Hamiltonian and uses only the existence of a single approximately local constant of motion. Our results do not provide any statement on the influence of the involved energy scale and are perfectly compatible with the existence of a dynamical mobility edge, in the sense of a zero-velocity Lieb-Robinson bound for low energy states [17,24]. So far, it seems unclear to what extend the transport investigated here can be related to energy or particle transport as investigated in the context of condensed matter systems. In fact, we would expect that such stronger form of transport is not possible in systems exhibiting many-body localisation.
As will be explained in the next section, the proof uses known equilibration results [10, 11] and thus establishes a recently proposed connection between transport and equilibration [3,23]. This implies two interesting features of our main theorem. First, the non-degenerate energy gaps are only needed to guarantee equilibration. Thus, the condition could be relaxed considerably accommodating some degeneracies in the model [11]. More importantly, the current literature on equilibration provides very little information on the involved time scales and the best known bounds are still exponentially slow in the system size [11]. This implies that, while we are able to prove information transport for systems exhibiting MBL, we obtain at the moment very little information on the time the information needs to travel through the system. Hence, we have chosen to state the results in terms of infinite time averages. We expect that MBL systems are indeed very slow to equilibrate, as indicated by the slow logarithmic growth of entanglement entropies [6].

Proof idea: Equilibration implies transport
Our results only rely on the existence of an approximately local constant of motion and assume no specific structure of the Hamiltonian. In order to present the argument in its simplest form, however, we will here assume that the Hamiltonian is diagonal and the local sites consist of spin-1/2 systems. A good example is the following Hamiltonian, which is the simplest model for MBL where σ z are the Pauli-Z-matrices and J i,j ∈ R decays exponentially with the distance between the spin. The proof of the full argument can be found in Appendix 3. It relies on perturbation arguments and a nifty construction of a local observable A, but the main idea is similar to the following proof sketch, which relies on using equilibration results following from non-degenerate energy gaps [10]. For this, we construct two objects. Firstly, a state that is the equal superposition of all eigenstates and secondly, a local operator A that initially has expectation value one with respect to this state, but at the same time has a zero diagonal in the energy eigenbasis and thus zero expectation value for the equilibrated infinite time average. Since equilibration guarantees that local expectation values are described by the infinite time average, this will allow us to conclude that the Heisenberg evolution of the operator A has to be non-local.

Lemma 1 (Diagonal Hamiltonians). Let H be a diagonal
Hamiltonian on a spin-1/2 lattice with non-degenerate eigenvalues and gaps. Let A = σ x j be the Pauli-X-matrix supported on spin j. Then H necessarily has transport on average in the sense that the operator A t has, on average, support outside any finite region S Proof. For the argument, we use a state vector |ψ that is initially a product with |+ on all sites. Let us point out that this |ψ simply reflects an equal superposition of all eigenstates of the system and is thus an infinite energy state. Since we assume that the Hamiltonian has non-degenerate energies, we know that the infinite time average of ρ = |ψ ψ| is diagonal, since all off-diagonal elements correspond to non-zero energy gaps and are thus dephased away. Moreover, as the diagonal is invariant under the time evolution, we know that the time-averaged state ω is the normalised identity matrix. Considering a subsystem S, we can use the non-degenerate energy gaps to employ known equilibration results [10] for the expected deviation from the time average Here N is the total number of spins and the effective dimension counts how many eigenstates of the Hamiltonian are part of the state The above result states that, for most times, the reduced state of ρ t looks like the identity. Due to the way equilibration is proven, the results also directly applies to the inverse evolution ρ −t [10].
To investigate the transport properties of the Hamiltonian, we look at the time evolution of an observable A consisting of a single Pauli-X-operator somewhere in the region S. The key trick is to use the initial expectation value and to insert time evolution operators Since we know that the equilibrated state is the normalised identity, the expectation value of any local traceless operator B has to vanish on average Since A 0 is traceless and the time-evolution leaves the trace invariant, we can conclude that the operator A t on average cannot be local anymore. More precisely, we have that where we used that | tr (Aρ)| ≤ A ρ 1 and have defined ρ S = tr S c (ρ). Next we use the inverse triangle inequality, Eq. () and insert 0 = tr(d −|S| I S tr S c (A)) which is using the fact that the reduced observable has zero expectation value with the infinite time average Another application of | tr (Aρ)| ≤ A ρ 1 allows us to use the equilibration results discussed previously. Using The main idea for the proof still can be carried out in the setting where the Hamiltonian is no longer assumed to be diagonal, but where only the existence of an approximately local constant of motion is guaranteed. For this, it is first assumed that the constant of motion is exactly local. This implies that it is possible to distinguish different sets of eigenvectors locally and thus allows to construct local observables that have zero diagonal in the eigenbasis of the Hamiltonian. Moreover, a state with large expectation value with respect to this observable can be constructed. This again allows to use the equilibration results for this state, together with the off-diagonality of the observable in order to prove transport. Finally, the argument is concluded by performing a perturbation analysis, which makes room for the decaying tails of approximately local constants of motion. The details of the argument are contained in Appendix 3. The same perturbation analysis can be carried out to construct an efficient spectral tensor network even in the case of approximately local constants of motion (Appendix 4, as recently proposed [7].

Discussion & outlook
In this work, we have shown that systems with suitably non-degenerate spectrum and approximately local constants of motion, necessarily have to have information transport. We explicitly construct local excitation operators whose effect spreads over arbitrary distances and thus giving rise to a protocol for using MBL systems for signalling. Our results are nicely complemented by recent work showing that if the Hamiltonian can be written in terms of approximately local constants of motion, logarithmic Lieb-Robinson bounds can be derived [7]. Our result can be seen as a rigorous proof that this logarithmic cone can never be tightened to a zero-velocity Lieb-Robinson bound, at least if one allows for infinite energy in the system. Let it be noted that, in contrast to the setting in Ref.
[7], we do not require any special decomposition of the Hamiltonian, but rather the existence of a single approximately local constant of motion is sufficient.
What is more, our work constitutes an important step towards proving transport in generic spin models and gives a sort of lower Lieb-Robinson bound for infinite times. As future work, it would surely be interesting to explore the precise form of the time-dependence, which naturally is linked to the ongoing debate on time scales of equilibration in local models. Any progress there would immediately imply a time scale for transport using the results contained in this work. Another important question is how the transport derived above is linked to the available energy scale in the system. In particular, it would be interesting how our results relate to the possibility of having a mobility edge and how they are connected to the presumed suppression of energy and particle transport in MBL systems.
Acknowledgements. We thank Christian Gogolin for many interesting discussions on possible links between equilibration and transport and Henrik Wilming and Tobias Osborne for numerous discussions on many-body localisation. We are grateful for support by the EU (SIQS, RAQUEL, AQuS, COST), the ERC (TAQ), the BMBF, and the Studienstiftung des Deutschen Volkes. WB is supported by the EPSRC.

Local systems
In this appendix, we review some basic definitions for local quantum many-body systems. We work with a finite lattice Λ with d-dimensional spin systems attached to each vertex of the lattice. We look at local regions X and denote by |X| the number of spins contained in such a region. For any fixed set X, we will introduce enlarged sets X l that contains X as well as all sites within distance l of X (Fig. 1). The distance measure will be the Manhatten metric, such that distances are always natural numbers. Local reductions of an observable will be performed by a map where S c denotes the complement of S. Due to the duality of operator and trace norm, we have Γ S (A) ≤ A for any region S. In the following, we introduce local operators and unitaries.
Definition 3 (Local observable). An operator A will be called (g, X)-local, if there exists a finite localisation region X such that for some function g : N → R with suitable decay in l.
Definition 4 (Local unitary). A unitary operator U will be called f -local, if the conjugation of a local observable A with localization region X remains local in the sense that for some function f : N → R with suitable decay in l.
Correspondingly, we will say that a Hamiltonian has flocal eigenvectors, if the unitary diagonalising it is f -local.
The main focus of this work is to investigate the transport of local operators. We work with the following definition of transport that captures spreading of support in the Heisenberg picture.
Definition 2 (Transport on average). We will say that a Hamiltonian has transport on average, if we can fix an error arbitrarily close to unity and there still exists a strictly local observable A with unit operator norm, such that, for each finite region S, truncation of the Heisenberg evolution to that finite region necessarily leads to this fixed error, as long as the system is large enough. More precisely, we have that Here · denotes the infinite time average.
The form of the estimate is well known for Lieb-Robinson bounds that provide an upper bound for the above quantity depending on time [22,25]. In this way, our definition can be seen as a "lower Lieb-Robinson bound" , albeit without any information on the corresponding time scale. Note that the above transport on average naturally also leads to transport for a large fraction of time maybe with respect to an adjusted error ε.
As in the case of Lieb-Robinson bounds [26], this definition can be connected to information propagation in the lattice. In particular, we can rewrite the reduction map Γ using an integration over unitaries drawn from the Haar measure Thus for systems showing transport according to Definition 2, we can find a state vector |ψ and such that a local excitation created by some local unitary V will be, on average, detectable at distances arbitrary far away In this sense, our definition of transport naturally captures the capability of a system to transport information.

Simple MBL model implies transport
In this appendix, we prove transport in two simple MBL models, one where the Hamiltonian is diagonal in the computational basis and one where the eigenstates are deformed by a f-local unitary (Definition 4).
We start with the following simple model which consists of interacting Pauli-Z-matrices with suitably random coefficients J i,j that decay with the distance between sites i and j. In this system, the eigenvectors are the computational basis and the energies and gaps will be non-degenerate due to the randomness in the model. For local dimension d > 2, one can easily extend the model, by allowing for coupling with arbitrary local and diagonal matrices. In this case, we further require a generalised Pauli-X-matrix on site j defined via matrix elements As discussed in the main text, this model implies transport in the following way.

Lemma 1 (Diagonal Hamiltonians). Let H be a diagonal Hamiltonian with non-degenerate eigenvalues and gaps. Let
A =σ x j be the generalised Pauli-X-matrix supported on spin j. Then H necessarily has transport on average in the sense that the operator A t has, on average, support outside any finite region S The proof is contained in the main text and directly carries over to the case with local dimension d > 2. Here, we will extent this result and show that the same construction can still be carried out in the case of approximately local eigenvectors, which are obtained from the computational basis by a joint f -local unitary (Def. 4).
Corollary 2 (f -local eigenvectors imply transport). Let H be a Hamiltonian with f -local eigenvectors and non-degenerate energies and gaps. Then H necessarily has transport on average in the sense that for any fixed finite region X l of diameter l, there exists a local operator A initially supported on X l with A = 1 such that A t has, on average, support outside any finite region S Proof. We will now use Lemma 1 in order to provide a proof for Corollary 2 . For this, we use that the Hamiltonian can be diagonalised by a f -local unitary V and work with the observable j is the generalised Pauli-X-matrix on some spin j within the set S. This operator will no longer be strictly local, but due to the f -locality of the unitary V , the operator can be truncated where X l denotes the set that contains the inital support, namely site j and all l-nearest neighbours. Here we used that the operator norm of A is one. From this, we can use the local reduction A l = Γ X l (A) as the local operator that will display transport. We will further need the time evolution of this truncated operator A l t = e itH A l e −itH , where we first truncate and then evolve it in time. Naturally the unitary time evolution does not change the norm difference. The proof relies on a series of triangle inequalities. First we use that for any state (9) Next we look at the two terms separately where we have inserted a zero term ± tr(A t ρ −t ) and have used the above truncation estimate in Eq.
( 2) . The other term can be estimated as follows Here we again inserted a zero term and used a norm estimate. To proceed, we insert one more zero term ± tr S Γ S (A t )d −|S| I S and use the triangle inequality, and use that Γ is a norm contractive map. These three terms can now easily be bounded. The first is small due to the flocality of the unitary V involved in constructing A, see Eq. 2 . The second term becomes small, once the time average is taken, which allows us to use known equilibration results () The third term vanishes completely, since the observable A has zeros on its diagonal. This completes the estimate of the second term in Eq. ( 2) Patching the estimates in Eqs. ( 2) and ( 2) together concludes the proof The above Hamiltonians are special instances of systems having local constants of motion. In the diagonal case, the constants of motion are simply the local Pauli-Z-matrices. Once they are deformed by a quasi-local unitary, exact locality is lost, but one still obtains a full set of approximately local constants of motion.

Constants of motion imply transport
In case the Hamiltonian has exactly local constants of motion [18] Z X supported on some region X, they can also be employed to obtain the operators A in the above construction. For this, let us assume that Z X = M k=1 λ k P k with exactly local projectors P k supported on X and M distinct eigenvalues. The goal is then to construct an operator that is block-off-diagonal with respect to the projectors P k . For this, let d min be the smallest dimension of the eigenspaces of Z X , when viewed as a local operator. For the construction, we fix two eigenspaces of Z X . The larger of the two is then truncated down to the dimension d trunc of the smaller one. Note that the resulting dimension of both spaces is lower bounded by d min . In these subspaces, we further fix some basis labelled by two indices |k, r where k labels the eigenspaces of Z X and r the basis vectors in each of these subspaces. We will denote the eigenspaces by k = 0 and k = 1. The operator A is constructed to be supported on the small region X and taken to be the flip operator between the subspaces A = dtrunc r |k = 0, r k = 1, r| + |k = 1, r k = 0, r| .
The operator norm of this observable is one and we will proceed by constructing an initial state that is an eigenstate of A to eigenvalue 1, but still has large effective dimension. For this, we pick the subspace with smaller dimension and take the equal superposition, denoted by |v of all eigenvectors in this subspace. For this, it is crucial to choose the subspace with smaller dimension, as the truncation in general, is not aligned with the eigenstates of the global Hamiltonian. The number of eigenvectors in the untruncated subspace will be lower bounded by d min = d min d N −|X| , which is simply the smallest eigenspace dimension of Z X when viewed as an operator on the full lattice. The initial state vector is then taken to be It is straightforward to check that this is indeed an eigenstate of A, since A 2 |v = |v . What is more, the state vector |ψ has an effective dimension lower bounded by d min . This gives the following result.
Corollary 3 (sIOM imply transport). Let H be a Hamiltonian with non-degenerate energies and gaps and Z X be a sIOM supported on X, with eigenspaces with dimension larger than d min . Then H necessarily has transport in the sense that for any finite region S containing X there exists a local operator A initially supported on X with A = 1 such that A t , on average, has support outside S Proof. To prove this statement, we can directly follow the proof of Lemma 1, as presented in the main text. Using the construction of the initial state and the observable A described above, we immediately obtain from the same equilibration results as before (Eq. ()). Inserting the effective dimension described above d eff ≥ d min = d min d N −|X| and d sys = d |S| concludes the proof.
In many localising systems, one does not expect the constants of motion to be strictly local, but only approximately local [18], meaning that Z is a (g, X)-local operator in the sense of Definition 3 . Using perturbation theory, it follows that this bound is sufficient to obtain local approximations for the eigenprojectors and makes it possible to once again construct an observable A that is transported through the system.
Theorem 1 (qIOM imply transport). Let H be a Hamiltonian with non-degenerate energies and gaps and Z be a qIOM with localisation region X, decay function g with spectral gap γ > 0 and eigenspaces with dimension larger than d min . Then H necessarily has transport on average in the sense that there exists a local operator A initially supported on X l ⊃ X with A = 1 such that A t , on average, has support outside any finite region S Let us remark that the first term can be chosen arbitrarily small by picking the initial support X l large enough and the second term decays exponentially with system size L, due to the growth of the degeneracy d min .
Note that the role of the eigenspace dimension d min is different to the previous corollary. Here, we look at the constant of motion as an operator on the full Hilbert space. We assume that the number of eigenvalues M and the spectral gap are independent of the system size. From this, perturbation theory can be used to show that the eigenspace dimension d min has to grow exponentially in the system size.
Proof. The first step of the proof is to show that the approximate locality of the constant of motion also implies quasilocal eigenprojectors. Let Z = M k=1 λ k P k and let γ denote the smallest spectral gap. Due to locality, we can express Z for each fixed l, as Let P l k be the eigenprojectors for the truncated observable. Perturbation theory assures us that the perturbed eigenspaces stay approximately orthogonal (Theorem VII.3.1 in Ref. [27]) which also implies Choosing the distance l large enough such that the function g becomes smaller than γ/2, we know that the perturbed and unperturbed eigenspaces have the same dimension [27]. This local approximation of the eigenprojectors of the constant of motion will be the basis for the construction of the observable A as well as the initial state ρ.
In order to construct the observable, we will work with the truncated constant of motion Γ X l (Z), fix two subspaces and construct the same flip operator as in the case of exactly local constants of motion A = dtrunc r |k = 0, r k = 1, r| + |k = 1, r k = 0, r| .
Without loss of generality, let k = 0 be the space with smaller dimension and k = 1 the one truncated to d trunc . Let P l 1 | I be the projector on the truncated subspace of P l 1 corresponding to the image of A.
For the initial state, we will use the corresponding subspaces, again labelled by k = 0, 1 of the full constant of motion Z. Again we pick the smaller of the two subspaces and define |v to be the equal superposition of all eigenstates within this space. The initial state vector is then By construction, the effective dimension and the equilibration results will be as in the case of a strictly local constant of motion. Crucial in the above construction is that we use the truncated constant of motion Γ X l (Z) for the observable A in order to ensure locality, while we use the full object Z for the initial state in order to achieve a large effective dimension. What remains to be shown is that despite this locality difference in the construction, we still achieve a large expectation value of A with |ψ , but an almost vanishing expectation value with the infinite time average.
As a first step, we will show that A is almost block-offdiagonal with respect to the eigenprojectors of the full constant of motion Z. Introducing the identity I = P l k + Q l k , this takes the following form Here we used that A is block-off-diagonal with respect to the truncated constant of motion Γ X l (Z). The same estimate holds for the projectors Q k . Using this, bounding the expectation value with the infinite time average is straightforward tr(Aω) = tr(AP 0 ωP 0 ) + tr(AQ 0 ωQ 0 ) We now have to show that the expectation value of A with ρ is large initially In the following, we will show that the first term is almost one, while the other two almost vanish due to the blockoff-diagonality. For the first term, we will use that A 2 = P l 0 + P l 1 | I , where P l 1 | I is the projector onto the image of A in P l 1 . Using that v| Q |v can only increase if we enlarge the subspace of the projector Q, we obtain where we used ( 3). The second term can be bounded directly using block-off-diagonality The last term, finally can be bounded as follows.
Putting together the estimates for the expectation value of A with the initial state, the equilibration result and the expectation value of A with the infinite time average, we obtain the desired bound. More precisely, we choose ρ = |ψ ψ| and proceed as follows Inserting the effective dimension d eff = d min and using Eqs.

Approximate spectral tensor networks
In Ref. [18], it is shown that if a Hamiltonian has suitable local constants of motion (sIOM), then each eigenprojector can be efficiently represented as a matrix product operator. Moreover, it is rigorously derived that there exists an efficient spectral tensor network for all eigenprojectors at the same time. Ref.
[18] then proceeds to sketch the case of approximately local constant of motion (qIOM), for which similar conclusions are reached. In this appendix, we show that indeed, even for approximately local constants of motion with robust spectrum (Definition 1), one can rigorously obtain a spectral tensor network.
Result (Efficient spectral tensor networks from qIOM). Let H be a Hamiltonian with an extensive number of approximately local constants of motion (Def. 1) with |X| ≤ L and g(l) ≤ c 1 exp(−c 2 l), for suitable constants c 1 , c 2 > 0. We assume that the qIOM are algebraically independent, commute with each other and have suitable distributed support on the lattice. Then there exists an efficient spectral tensor network representation for all eigenprojectors of H.
The proof of this statement directly follows from Ref.
[18], together with our Corollary 1. We start from the observation that the approximate locality of the constant of motion also implies quasi-local eigenprojectors, in the sense that and that the perturbed and unperturbed eigenspaces have the same dimension. Using this approximation, one finds that projectors onto a qIOM eigenspace can be efficiently approximated by matrix-product operators. For a given site j, call A the subset of sites for which the MPO approximations have a support that includes j. With the same argument as in Ref.
[18], choosing a path in the supports of the sIOM and performing singular value decompositions, as outlined in Ref.
[18], one finds that the collection of all qIOM in A can again be written as a matrix-product operator. The stability Lemma 2 below concludes the proof.
Lemma 2 (Stability). Let {Z j } be a set of N qIOMs with a lower uniform bound γ > 0 on their minimal spectral gaps and uniform upper bounds L and g(l) on size and decay of their localisation regions X j , such that Z j − Γ X l j ⊃Xj (Z j ) ≤ g(l) for any X l j containing X j together with a buffer region of size l. Then if P j,m denotes the eigenprojector of Z j for eigenvalue m we have P j1,m1 · · · P j N ,mn − P l j1,m1 · · · P l j N ,N1 ≤ 2N g(l) γ with P l ji,mi = Γ X l j (P ji,mi ) being strictly local. Proof. The proof utilises perturbation theory on the level of the single eigenprojectors P l ji,mi similar to the proof of Corollary 1. Using the triangle inequality we can upper bound the norm difference as P j1,m1 · · · P j N ,mn − P l j1,m1 · · · P l j N ,N1 ≤ N k=1 P j k ,m k − P l j k ,m k .
The result now follows from Eq. ( 3) and our uniformity assumptions.