Quantum Hamiltonian complexity and the detectability lemma
Local Hamiltonians, the central object of study in condensed matter physics, are the quantum analogue of CSPs, and ground states of Hamiltonians are the quantum analogue of satisfying assignments. The major difference between the two is the existence of multi-particle entanglement in the ground state, which introduces a whole new level of difficulty in tackling questions such as quantum PCP, quantum analogues of amplification, etc.
The Lieb-Robinson bound is a sophisticated analytic tool used in condensed matter physics for handling quantum correlations in ground states, by bounding the velocity at which disturbances propagate in quantum local systems. In this paper we show that the detectability lemma (introduced in a different context in LABEL:ref:Aha09b), when viewed from the right perspective, can be used in place of the Lieb-Robinson bound for the rich case of frustration free Hamiltonians. The advantage of this is that the resulting proofs are simpler and more combinatorial, and may be generalizable to solve some of the most fundamental questions in Hamiltonian complexity. Additionally, we give an alternative proof of the detectability lemma, which is not only simple and intuitive, but also removes a key restriction in the original statement, making it more suitable for this new context.
Specifically, we use the detectability lemma to give a simpler proof of Hastings’ seminal 1D area law  for frustration-free systems. Proving the area law for two and higher dimensions is one of the most important open questions in Hamiltonian complexity, and the combinatorial nature of the detectability lemma based proof and the resulting simplification holds out hope for a possible generalization. We also provide a one page proof of Hastings’ proof that the correlations in the ground states of gapped Hamiltonians decay exponentially with the distance (once again, restricted to frustration-free systems). We argue that the detectability lemma in this form constitutes a basic tool for the study of local Hamiltonians and their ground states from a computational point of view.
Local Hamiltonians and ground states, the central object of study of condensed matter physics, are the quantum analogues of the central objects of study in computational complexity: constraint satisfaction problems (CSP) and their satisfying assignments. This connection, which ties together two seemingly very different areas, is the starting point for the emergence of the new field, Quantum Hamiltonian Complexity, in which properties of local Hamiltonians and ground states are being studied from a computational complexity point of view. Over the past few years, this direction has shed exciting new insights into quantum information theory as well as into quantum physics. Of crucial importance here is the difference between the quantum and classical domains: the quantum analogue of the satisfying assignment, namely the ground state, can exhibit extremely intricate multi-particle entanglement. This additional player in the game makes borrowing results from the classical domain to the quantum domain extremely challenging, cf. the wide open major open problem of whether a quantum analogue of PCP holds ; it also opens up completely new directions of research regarding the entanglement properties of ground states of local Hamiltonians.
General quantum states require complex numbers to describe. One of the major goals of quantum Hamiltonian complexity is to derive bounds on the entanglement exhibited in ground states of interesting classes of local Hamiltonians; the purpose of those bounds and restrictions on the entanglement is to lead to an efficient description and analysis of ground states in cases of interest. There is a beautiful sequence of papers using structures called tensor networks, with special cases such as MPS [3, 4, 5, 6], PEPS , TN , and MERA , which provide such efficient descriptions in certain cases.
Area laws constitute one of the most important tools for bounding entanglement in such systems. Consider the interaction graph (hypergraph) associated with a local Hamiltonian – it has a vertex for each particle and an edge for each term of the Hamiltonian. Intuitively and very roughly, an area law says that entanglement is local in this interaction graph in the following sense: consider a subset of particles . Then the entanglement between and in the ground state is locally “concentrated” along the edges between and ; more precisely, the area law states that the entanglement entropy across the cut is big-Oh of the number of edges crossing between and . This is clearly a very strong restriction on the entropy, which in the general case would be of order of the number of particles (nodes) in . Proving area laws for typical classes of Hamiltonians is thus a holy grail in quantum Hamiltonian complexity.
A few years ago, in a seminal paper , Hastings proved that the area law holds for 1D systems (i.e., when the interaction graph is a path), for gapped Hamiltonian – that is, Hamiltonians whose overall spectral gap is of order . In this case, the area law says that ground state entanglement across any contiguous cut is bounded by a constant. From this, one can deduce that the ground state of such systems can be described efficiently (by an MPS of polynomial bond dimension – see LABEL:ref:Has07). The question of whether ground states in two and higher dimensions obey an area law is still wide open.
Hastings’ proof of the 1D area law, and many other proofs related to entanglement and correlations in ground states, use sophisticated analytic methods. Perhaps the most important of those is the famous Lieb-Robinson bound (LR bound) [10, 11], which bounds the velocity at which disturbances propagate in quantum local systems; Fourier analysis, and other techniques are important players too. These analytic tools constitute a major barrier for a fuller participation by computer scientists in this important aspect of Hamiltonian complexity. Also, these analytic techniques seem to inherently involve the dynamics of the system in time, according to the Hamiltonian. However, purely from an aesthetics point of view, it should be possible to explain kinematic results about the ground state without resorting to dynamical arguments (which is what the LR bound is). Or, in other words, without adding the extra dimension of time to the problem. In addition, the kinematic problem seems, at least on the surface, to be of a combinatorial nature, thereby suggesting a combinatorial solution.
In this paper we introduce a combinatorial tool to tackle the above mentioned problems, and, in particular, to get a handle on correlations and entanglement in ground states of local Hamiltonians. This is a simple, basic version of the detectability lemma of LABEL:ref:Aha09b. We demonstrate that when the system is frustration-free, many of the results that rely on the traditional analytic tools can be obtained in a much simpler, direct and intuitive way using this tool; we argue that the detectability lemma in this form constitutes a basic tool for the study of local Hamiltonians and their ground states from a computational point of view.
Our starting point is the Detectability Lemma (DL) introduced in LABEL:ref:Aha09b. There, the motivation for the DL was quite specific: to help translate classical results about CSPs to quantum results about local Hamiltonians. It was used to prove a quantum analog of gap amplification (a component of Dinur’s proof of the PCP theorem ). The DL made it possible to sensibly make a statement of the form “If the ground state energy is at least then the probability that it violates at least terms of the Hamiltonian is bounded below by a constant”. The DL of LABEL:ref:Aha09b holds under the mild assumption (which is essentially true in most interesting cases) that each particle participates in a bounded number of terms of the Hamiltonian, and therefore the terms of the Hamiltonian can be partitioned into a constant number of layers, each consisting of terms acting on disjoint sets of particles. LABEL:ref:Aha09b also required an additional technical assumption, that the number of distinct types of terms of the Hamiltonian are bounded.
Here, we reformulate the DL and put it in a much broader and basic context. Our reformulation of the DL asks the following question: consider a gapped frustration-free local Hamiltonian with . i.e., the ground energy of is , and the spectral gap is . The frustration-free assumption means that the ground state minimizes the energy of every local term, so no term is “frustrated”. Can we approximate the projection on the ground state, , by a “local” operator? Such a local approximation would be extremely useful, as it would enable deducing local properties of the ground states such as area laws and decay of correlations. Indeed, such an approximation of a projection on the ground state is essentially what is done by the traditional analytic tools that use the LR bound, as we explain in Sec. 4. The approximation offered by the DL, however, has more of a combinatorial flavor, and is therefore much easier to handle.
A natural first guess of such a local approximation of is the positive semi-definite operator , where is the number of terms in the Hamiltonian. fixes the ground state, and shrinks all the orthogonal space to it by a factor; however the shrinkage is very limited, by a factor of . To get a good approximation, one would need to apply this operator polynomially many times, and by this we would lose the locality of the operator. Indeed, the expression contains products of overlapping terms whose overall support is of the order the size of the system. Our challenge is therefore to get a local operator that preserves the ground state but shrinks the orthogonal subspace by a constant factor, rather than by . For simplicity of presentation in the introduction, let us consider the simplest scenario, in which the particles are set on a 1D chain, and the interactions are two local. Denote by the projection on the ground state of the terms . Notice that the terms in the Hamiltonian can be partitioned into two layers, the even and odd terms, each acting on disjoint sets of particles (see Fig. 1);
Denote by the product of the projections on the ground spaces of all odd terms and by the product for the even terms. Then the operator is the “local” operator we want. The DL states:
Lemma 1.1 ( Detectability Lemma (DL) in )
Let , and let be the orthogonal complement of the ground space. Then
The DL says that the application of to any vector moves the vector closer to the ground state of by cutting down the mass in the orthogonal subspace by a constant factor. This implies that , the projection into the ground space of , can be approximated to within exponentially good precision by applying the operator times: .
Let us explain why this operator is indeed “local”. When is applied times to some local perturbation that acts on the ground state , there is a pyramid-shaped “causality cone” of projections that is defined by . These are simply all terms which are graph-connected to the operator (see Fig. 2). All the projections outside that cone commute with and can therefore be absorbed in the ground state (since ), leaving us with a local operator of support of size . Effectively, acts non-trivially only on a region of width , when applied to .
We give here a new simple proof of this reformulation of the DL, in the process dropping the assumption of LABEL:ref:Aha09b about the number of distinct types of terms of the Hamiltonian.
The proof hinges on the following observation, which we refer to as the norm-energy trade off. Assume by contradiction that does not move a vector , which is orthogonal to the ground state, very much. Then must be very close to the range of each , but since the range of is the null space of the local term , this means that the energy must be small. However, on the other hand, the sum of those energies must be larger than , since is orthogonal to the ground space; this implies that the shrinkage must be quite significant, providing an upper bound on the norm of the vector .
However, the above argument is not sufficiently strong. Since there are terms , the energy contribution of each term can be as small as ; this will lead to a factor of in the lemma, which is not strong enough. The key point is that the energy-norm trade off can be applied locally, using the tensorial structure of ; we break the movement of to into disjoint sequential steps and then relate the contributions to the energy of each of the terms with the shrinkage resulting from each step; one might suspect that entanglement could prevent such an analysis in which shrinkage accumulates but the point is exactly that the local structure of the problem allows this accumulation to happen. Think very simplistically of the state subjected to the local terms . A projection of this state on the ground state of for any one of qubits , results in a shrinkage by a factor of , but once one projection is applied in one location, the shrinkage is exhausted and no more shrinkage is to be gained by a projection in another location. That this entanglement related phenomenon does not happen in the DL scenario is due to the locality of the operators involved; it highlights that the way the state can be entangled is severely limited.
We demonstrate the applicability of this reformulation of the DL by providing significant simplifications of the proof of Hastings’ area law in 1D , using the DL in two key points, bypassing completely the analytic methods. By this we hope to make this important result accessible to a wider audience, as well as possibly extendable to higher dimensions. The outline of the proof still follows that of Hastings, but now becomes much easier to understand; we defer the explanation of how the proof goes and how the detectability lemma enters the picture to Sec. 5.
To give another example, we provide a one page, very simple proof of Hastings’ celebrated result that the correlations in the ground states of gapped Hamiltonians decay exponentially with the distance . Unlike the area law, this applies to -dimensional grids for any constant . More precisely, consider two observables and that are local and act on sets of particles that are of distance on the grid; the decay of correlations means that the expectation value of their product is almost as that of the product of their expectation, up to an error which decays exponentially in .
We mention that at first sight, one might connect the exponential decay of correlations to an intuition that entanglement between a region and its surrounding is “located” only close to the boundary of , and thus scales like the area rather than like the volume. Though an appealing intuition, such an implication of exponential decay of correlation to area laws is not known, and indeed quantum expanders provide a counter-example to such a naïve connection .
In both of those proofs, the DL replaces a combination of the Lieb-Robinson bound with other analytic tools; this works of course only when the DL is applicable, namely, for the rich case of frustration-free Hamiltonians. The restriction to frustration-free Hamiltonians may seem quite strong. We note, however, that there are various frustration-free systems that are interesting from a physics and a computational points of view, such as the ferromagnetic XXZ model, the AKLT model , and stabilizer codes such as the Toric code . In addition, many of the quantum phenomenon in quantum Hamiltonian complexity are revealed already in the context of frustration-free Hamiltonians, and the major open problems in this area (e.g., quantum PCP and 2D area law) are wide open already for this case. Much is to be learned from studying frustration-free Hamiltonians, before we proceed to the more general case; it seems that the simpler combinatorial nature of the DL in this case might provide a new handle to those questions, and there are reasons to believe that a proof of an area law for frustration-free systems might be extendable to the general case.
To illustrate how exactly the is DL related to the analytic methods, we start our more technical discussion with a toy application comparing the usage of the LR bound to the alternative route offered by the DL, in Sec. 4.
Related work and further directions:
The DL seems to be connected to various diverse scientific areas. The connections to the LR bound and other analytic tools used in condensed matter physics are discussed extensively in Sec. 4; one other connection is to view of the DL operator as a special instance of the general Method of Alternating Projections (MAP), that was first studied by von Neumann . In that method one applies a fixed sequence of projections in order to approach the intersection subspace. In the general setting, the projections are not assumed to be local, nor the Hilbert space is assumed to be of finite dimension. In recent results , the convergence rate is given as a function of the Fridriechs angle, which is not easily related to a physical quantity. The DL, on the other hand, is a MAP under the special assumption that the projections are local, associated with a frustration-free -local Hamiltonian, with a convergence rate that is given as a function of the spectral gap. It would be interesting to see if more insight can be derived from these connections.
Recently, much attention was given to a quantum algorithm which, given a local Hamiltonian, uses a process involving random measurements of the energies of the local terms to approach the ground state efficiently (for certain cases) [18, 19]. The algorithm discussed in those papers carries similarities to the situation we are handling here, despite the fact that measurements are applied rather than projections, and also that the terms are chosen randomly, rather than in some fixed order. It seems that the DL lemma, and the energy-norm trade off, could potentially be useful also for the analysis of such algorithms. In particular, it would be very interesting to see a version of the detectability lemma which applies for the case in which the terms are chosen randomly.
As discussed above, it is a wide open question to apply the combinatorial tools presented in this paper to the major open problems of quantum PCP and area laws in dimensions higher than , as well as to many other basic open questions in quantum Hamiltonian complexity.
We start with notations and preliminaries in Sec. 2, and then proceed to the statement and proof of the DL in Sec. 3. In Sec. 4, we provide the example comparing the LR bound approach to the DL one. We then proceed to the area law proof in Sec. 5, and conclude with the one page proof of the exponential decay in Sec. 6.
2 Notations and Preliminaries
We consider a -local Hamiltonian acting on , the space of particles of dimension . where each is a non-negative and bounded operator that acts non-trivially on a constant number of qubits (hence the term local Hamiltonian). We assume that has a ground space of energy , which must therefore also be a common zero eigenspace of all terms . This means that is frustration free. We also assume that is “gapped”, meaning that its lowest eigenvalue is 0 (the ground energy) and all the next are equal or larger than some constant . We denote by the orthogonal complement ground space of . Thus is an invariant subspace for , and
Most of these assumptions, except for perhaps the frustration-free assumption, are very often used in condensed matter physics.
Throughout this paper we further assume that the ’s are projections, and hence would be denoted by . We define to be the projection on the ground space of , . The assumption that is made of projections is not actually a restriction because we can reduce any frustration-free, bounded and gapped system into that case. Specifically, for with , and a spectral gap , we first add an appropriate constant to each such that their ground energy is 0. Then for every we define as the projection into the space where the energy of is greater than 0 and as the projection to the ground space of . Finally, we define the auxiliary Hamiltonian . This system is frustration free because the original ground states would also be ground states in with a vanishing energy. Moreover, for any state and every ,
and therefore the gap in is . It follows that all of our results can be applied to bounded frustration-free Hamiltonians by replacing the gap in DL with the scaled version .
Given a state and a partition of the qubits to two non intersecting sets, and , with corresponding Hilbert spaces , we can consider the Schmidt decomposition of the state along this cut: . Here are the Schmidt coefficients. Their squares are equal to the non-zero eigenvalues of the reduced density matrices to either side of the cut and , which we denote by . The Schmidt rank of is then the number of non-zero eigenvalues (or Schmidt coefficients ), and the entanglement entropy is the entropy of the set , or, equivalently, the von Neumann entropy of the matrix . A straightforward corollary of the Eckart-Young theorem  is then that the truncated Schmidt decomposition provides the best approximation to a vector in the following sense:
Let be a vector on , and let be the eigenvalues of its reduced density matrix. The largest inner product between and a norm one vector with Schmidt rank is .
3 The detectability lemma: A new proof
For clarity of presentation, we will prove the DL in the case stated in the introduction: where the particles are set on a line and the local terms are two-local involving nearest neighbors. This proof contains all the necessary ingredients for the proof of the more general DL in the case where the Hamiltonian has local terms that can be partitioned into layers; we make the precise statement of the more general case at the end of this section.
We begin with a simple lemma that quantifies the norm-energy trade-off in the simple case of two projections : we show that if the application of does not move a vector very much then the energy of that vector with respect to must be small:
Given arbitrary projections and of norm 1, if then
The proof is given in the Appendix. Let us now proceed to prove the detectability lemma.
Suppose is a norm 1 state that is orthogonal to the ground space, and define . Notice that for every ground state , and so is orthogonal to the ground space. We would like to show that
We will find both a lower and upper bounds for the energy of , , which will give us an inequality for , from which Eq. (4) will follow.
The lower bound is straightforward since is orthogonal to the ground state, and so
We shall now upper bound the energy by carefully upper bounding the contributions of the individual terms . We begin by noting that these terms are equal to for odd since and for any odd (recall that are products of the projections ). We now want to bound the contributions coming from the even terms.
For this purpose we present in a convenient form, by reordering its terms. We call the triplet product of projections pyramids, and denote them by ; The remaining terms are combined to the operator . See Fig. 3 for an illustration of this structure in 1D. Notice that by just using the fact that and commute when and are not consecutive, we can write:
where is the number of pyramids which is approximately .
We will use this reordering to bound the energy contribution of the terms ; a symmetric argument will bound the remaining even terms etc. The energy contribution of will be related to the amount of movement produced by the portion of of the operator .
The key point in providing this bound is this. We view the transformation of as a series of steps given by the application of the pyramids . Specifically, letting , we consider the transformation . The square of the norm of the first state, after applying , is . Let be the “shrinkage” of the square norm (or movement) resulting from the application of the th pyramid, for .
We shall now prove, using Lemma 3.1, that the shrinkage is related to the energy of the operator at the top of the same pyramid :
and recall that . We can therefore apply Lemma 3.1 to (with and ), and conclude that . Consequently .
Using this upper bound gives an upper bound for the energy contribution for , :
with the constraint on the norm . The right hand side is maximized when all the s are equal to each other, i.e., , and therefore we are left with an upper bound of the energy coming from as:
where the last inequality follows from the fact
1that for every , we have .
For the energy of , a similar decomposition to can be made, thereby upper bounding the energy by . Combining the energy upper and lower bounds we therefore get
The above proof can be easily generalized to other geometries. In the general case, in accordance with Sec. 2, we assume we have a -local, frustration-free Hamiltonian that is made of projections and has a spectral gap . We further assume that each particle participates in a constant number of projections, and therefore the can be partitioned into a constant number of layers; each layer is made of projections that do not intersect each other and are therefore commuting.
Then for each layer we define the projection as the product of all that are in the layer, and define the DL operator by
Finally, we define to be the number of sets of pyramids that are necessary to estimate the energy contribution of all the terms. In the 1D case that we proved, we had , because only the even layer contributed energy and we needed two sets of pyramids to cover that layer. In the general case it is easy to see that can be crudely bounded by .
Using the above definitions, the general DL is
Lemma 3.2 (The detectability lemma)
Consider the local Hamiltonian system that is described above. Then
4 Comparing the Lieb-Robinson bound approach and the detectability lemma approach
In this section, we compare the DL with a standard method used in many of the seminal results in quantum Hamiltonian complexity, such as Hastings’ areas law for 1D gapped systems , and Hastings’ exponential decay of correlations proof . The method combines the use of the Lieb-Robinson bound (LR bound), with a Fourier analysis and the existence of a gap, to reveal the locality properties of the ground state. More specifically, the method uses these tools to approximate expressions that involve the projection operator to the ground state, , by local operators.
To understand how this is done, let us concentrate on a simple example of locality in the ground state, and derive it using both the DL and the LR bound.
We focus on , the projection on the ground space of . On the surface, this projector seems very far from being local in any sense. Nevertheless, in gapped systems it does possess some locality properties that are crucial to the analysis of correlations and entanglement in the ground state. To see this, one standartly considers an approximation of by another operator:
Here is a free parameter to be chosen as appropriate. For an eigenvector of with eigenvalue , we have . Consequently, if the system has a constant spectral gap , indeed approximates well:
We now want to argue regarding the local nature of in various contexts. Let us illustrate the LR bound approach with a simple example: consider the expression , where is a ground state of the system with zero energy, and is some local perturbation. It is easy to see that
is the time evolution of the perturbation . The key point is now to use the famous LR bound, to approximate it by a “local” operator, i.e., an operator which acts only on the “neighborhood” of the particles on which acts. The following is an immediate corollary of the original LR bound, which we omit for sake of brevity. The full statement of the LR bound, together with the proof of this corollary can be found in LABEL:ref:Has10.
Theorem 4.1 (Lieb-Robinson bound (LR bound) , adapted )
Given a local Hamiltonian on particles, there exists a constant velocity s.t. can be approximated by an operator denoted whose support is inside a ball of radius around the support of , s.t.
Given a length scale , we may now set in Eq. (12) and obtain
In the above series of approximations implies an approximation of up to an error of . The 1st approximation follows from the assumption of the constant gap and Eq. (11). The 2nd approximation is due to the exponential decay of the filter function , and the 3rd is due to the LR bound. We therefore get an exponentially (in ) good approximation to in the expression by an operator which is -local.
Let us now derive the same result for the frustration-free case using the DL. First, we approximate the ground space projection by applying the DL operator for times. By Eq. (10), leaves the ground space invariant while shrinking the orthogonal space by a constant factor. Therefore
We now write
and consider the expression . By assumption, the system is frustration free, and therefore every local projection operator that appears in leaves invariant: . We now consider the “causality cone” of projections in that are defined by . These are simply the projections that are graph-connected to when all the projections in are arranged in consecutive layers (see Fig. 2). The main observation is that all the projections outside this causality cone commute with , and can therefore be absorbed by . We are therefore left only with the projections of the causality cone, whose support size is proportional to . In other words, just as in the LR bound method, we found an exponentially (in ) good approximation to in the expression by an operator which is -local.
This kind of reasoning, with appropriate modifications, is used in both the 1D area-law and the exponential decay of correlations, that are presented in the following sections.
5 The area law in 1D using the detectability lemma
Throughout this section, we let be a 2-local frustration-free 1D Hamiltonian that is made of projections acting on particles of dimension . Assume that has a unique ground state and a spectral gap , and set in accordance with the shrinking exponent of the DL, in Eq. (1). We notice that in the interesting limit , we have , and generally, for for , have . In order to keep the presentation simple, we shall assume throughout this section that , and prove the following version of a one dimensional area law:
Theorem 5.1 (Area Law for frustration free Hamiltonians in )
For any contiguous cut along the chain, the entanglement entropy of the ground state across the cut is bounded by a constant which depends on the dimensionality of the particles and on the spectral gap ; specifically,
The proof relies on two main lemmas. The first shows that for any cut along the line, there is a product state that has a constant inner product with the ground state:
Lemma 5.2 (Constant overlap with a product state)
For every cut, there is a product state such that , with .
The second lemma shows that if there exists a product state with a constant overlap with the ground state , then has finite entanglement entropy:
Lemma 5.3 (Constant overlap with a product state implies finite entropy)
If for some cut there exists a product state such that , then the entanglement entropy of across the cut is bounded by
5.1 Constant overlap implies finite entropy (proof of Lemma 5.3)
In this section we prove Lemma 5.3. The DL is clearly the right tool for the task, since it provides a “local” operator that can be repeatedly applied to the promised product state without increasing its entanglement rank much, while exponentially decreasing its distance from the ground state.
The only thing that is not entirely clear is how to get a constant bound on the entanglement entropy of the ground state, since a straightforward argument would mean applying the operator non-constant number of times to get arbitrarily close to the ground state. The key is to observe that after applications of the DL we get a state with a bounded Schmidt rank that is close to the ground state, and by Fact 2.1, this gives us a bound on the sum of the largest Schmidt coefficients of the ground state. With these bounds we can find a pessimistic constant upper bound on the entanglement entropy. We can now proceed to the more detailed proof.
Consider then a cut in the line between the particles and , and let be the local term in that involves . Assume that along that cut, the product state has a constant projection on the ground state :
where , and . We now apply the operator DL operator times on . We obtain
where and . Let be the normalized version of . Then
This means are exponentially close to the ground state, as a function of .
How entangled are those states?
We notice that at each application of , the entanglement
rank of the state can only increase by a multiplicative factor of
: for every , the projection term in
works entirely left to the cut or entirely right to the cut, thereby
not increasing the Schmidt rank of the state. The only projection in
that may increase the rank is , and as it is a 2-local
projection that works on -dimensional particles, it can at most
increase it by a factor of .
We have therefore obtained a family of states with Schmidt ranks bounded by , which are closer and closer to . Then using Fact 2.1, together with Eq. (18), it follows that the eigenvalues of the reduced density matrix of along the cut, , must satisfy
which implies the following series of inequalities:
From here, the desired upper bound on the entropy can be deduced by choosing the distribution of maximal entropy which still satisfies the inequalities in Eq. (21). The following lemma, whose proof can be found in Appendix B, gives one such bound:
Consider a probability distribution whose values are ordered in a non-increasing fashion, , and let be an integer and some constants such that
Then the entropy of is upper bounded by
Substituting and gives
But and , and so
where the last inequality follows from the assumption that .
5.2 A product state having constant overlap with (proof of Lemma 5.2)
The obvious candidate for a tensor product state with a constant overlap with is the mixed state , where is the reduced density matrix of to the left of the cut, and is the reduced density matrix to the right.
Let us assume for contradiction that the overlap between and , and in fact with any tensor product state along a certain cut, is less than for some sufficiently large constant . If the overlap is small, then there is a measurement that distinguishes from with probability of at least ; this is simply the projection on the ground state, .
The challenge is to show that there is a local such measurement, i.e., a measurement confined to a local window, which distinguishes these two states almost as well. Using the DL we shall now find such local measurement that distinguishes with a slightly worse probability .
Let us denote by (respectively ) the reduced density matrix of restricted to the particles to the left (respectively right) of the cut. Also, let be restricted to the particles, on each side of the cut. We refer to the state as the “disentangled” version of the state . The following lemma shows that under the assumption that has low overlap with every product state (along a given cut), there exists a measurement confined to the window of particles around the cut, that with high probability distinguishes from .
Lemma 5.5 (Existence of a distinguishing measurement)
Assuming that the overlap of the ground state with any product state satisfies , there is a measurement that distinguishes from with probability .
The DL ensures that by applying the layers one by one, we converge to the projection on the ground state quickly, and it is this projection that is exactly the distinguishing measurement we want to approximate. We can thus apply only times, approximating the projection on the ground space; now, following the intuition explained in the introduction (and in the example of Sec. 4), only the causality cone of the cut should be used in this measurement, and the rest of the operators in those layers are swallowed by the state being measured; this amounts to a measurement which is restricted to the interval and still distinguishes well enough. The detailed proof can be found in the appendix.
The fact that such a measurement exists, distinguishing the original state confined to the window from its “disentangled” version, with high probability, must somehow indicate that there is a lot of entanglement along the cut, whose disentanglement caused this distinguishability. This can be made precise using an information-theoretical argument:
Lemma 5.6 (Distinguishing measurement implies large difference in entropies)
If there is a measurement that distinguishes from with probability of at least , then
The lemma implies that the entropy in is significantly larger than , implying that disentangling along the cut has introduced a lot of new entropy. The proof is simple, based on relative entropy; essentially, all it uses is the fact that a measurement that distinguishes with high probability two states, implies high relative entropy between the results of the measurements. Once again, details can be found in the appendix.
To finish the proof of Lemma 5.2, we now need to derive a contradiction. Denote by the value of , with being the segment centered around the cut that provides our contradictory assumption (namely, that any tensor product state has less than inner product with the ground state). Under these conditions, Lemma 5.5 applies, and hence also the conditions of Lemma 5.6 apply to this segment. Applying Lemma 5.6 we conclude that . We now want to recursively apply this inequality, for the long segments on both sides of the cut, and then for the -long segments within those segments, and so on. The problem is that the cuts now move to different locations within the long window, and so our assumption no longer applies for these cuts. However, if the inner product state with any tensor product state is small along the original cut, it can be shown to be quite small also along near-by cuts, and so all the above arguments can be applied for those cuts too. This can be formalized in the following claim, whose easy proof can once again be found in the appendix:
If for all product states across cut then for all product states across any cut with .
We therefore assume by contradiction that the inner product along a given cut is smaller than , such that , and so along all the cuts in the window we have that the inner product of the states is at most , and hence our assumptions apply. We can therefore use the same argument recursively. Since , we get (for a power of 2), . Choosing such that makes thus giving a contradiction. Using the fact that , this can be achieved by since .
6 Exponential decay of correlations
Consider now a Hamiltonian which is -local and set on a -dimensional grid. Once again, we assume that are projections, and that is frustration free with a unique ground state (i.e., ), and a spectral gap . We wish to show
Decay of Correlations in ground states of gapped Hamiltonians on a -Dim grid:
Consider a setting as described above. Let be two local observables whose distance on the grid from each other is . Denote , . Then
Let us now consider two operators: : is defined by applying the DL times to and discarding all projections outside the causality cone of . is chosen such that the resulting cone will not overlap with (see Fig. 4). Therefore , with the proportionality constant that is a geometrical factor. is the complement of , i.e., it the layers that one get by applying the DL times, but with a “hole” where the causality cone of is. Together, we have – See Fig. 4 for an illustration in 1D.
leave the ground-state invariant. In addition, they commute with and respectively, hence
Assuming that the ground state is unique, , and therefore
We are grateful to Matt Hastings, Tobias Osborne and Bruno Nachtergaele for inspiring discussions about the above and related topics.
Itai Arad acknowledges support by Julia Kempe’s ERC Starting Grant QUCO and Julia Kempe’s Individual Research Grant of the Israeli Science Foundation 759/07.
Appendix A Proof of Norm-Energy trade-off, Lemma 3.1
Set . Then
By definition, is a normalized vector inside the support of and therefore for every vector , we have . Plugging this to the equality above, we find
where the last inequality follows from the fact that .
Appendix B Upper bound on the Entropy
Call the set of weights for the ’th block. Then the constraints in Eq. (20) imply that for every block ,