An area law for the entropy of low-energy states
It is often observed in the ground state of quantum lattice systems with local interactions that the entropy of a large region is proportional to its surface area. In some cases, this area law is corrected with a logarithmic factor. This contrasts with the fact that in almost all states of the Hilbert space, the entropy of a region is proportional to its volume. This paper shows that low-energy states have (at most) an area law with the logarithmic correction, provided two conditions hold: (i) the state has sufficient decay of correlations, (ii) the number of eigenstates with vanishing energy-density is not exponential in the volume. These two conditions are satisfied by many relevant systems. The central idea of the argument is that energy fluctuations inside a region can be observed by measuring the exterior and a superficial shell of the region.
Entropy quantifies the uncertainty about the state of a physical system. A bipartite system in a pure state has zero entropy, but the reduced state of one subsystem may have positive entropy. This is due to quantum correlations between the two subsystems, the entanglement. In fact, this entropy quantifies the entanglement in the sense of quantum information theory S ent .
In classical physics, the entropy of a region inside a spatially-extended system at finite temperature is proportional to the volume of the region—entropy is an extensive quantity. At zero temperature, it is small and independent of the region. In quantum physics, at finite temperature, the entropy of a region is also proportional to the volume. But it has been observed in several models that, at zero temperature, the entropy of a region is proportional to its surface area 80s ; Srednicki ; Audenaert ; Vidal ; Botero ; Jin ; Calabrese ; Plenio ; Casini . In some models of critical free fermions the entropy scales as the area times the logarithm of the volume Wolf ; Gioev . This has been presented as a violation of the area law, although the dimensionality of the scaling is still that of the area. A celebrated proof shows that any one-dimensional system with finite-range interactions and an energy gap above the ground state obeys a strict area law 1D .
The original motivation for this problem is the analogy with black-hole physics, where the thermodynamic entropy is proportional to the surface area of the event horizon Bekenstein ; 80s ; Srednicki . The second motivation is to guide the development of efficient methods for simulating quantum systems with classical computers. The number of parameters needed for specifying an arbitrary pure state of an -partite system is exponential in . If the state is not entangled, the number of parameters is proportional to . Hence, there seems to be a correspondence between entanglement and complexity. In one spatial dimension, the relation between entropy and the complexity of simulating a system is well understood V ; Schuch ; Vidal . The third motivation is to understand the kind of states that arise in quantum many-body systems with strong interactions. Almost all states in the Hilbert space obey a volume law for the entropy Hayden . Hence, area laws tell a lot about the multipartite entanglement structure. At a finer level, the specific form of an area law tells additional information about the system: the logarithmic correction is a signature of criticality Audenaert ; Vidal ; Wolf ; Gioev ; Calabrese ; and the appearance of a negative constant is a signature of topological order topo . For further overview of the topics around area laws see the review article cited review .
Ii Results and summary
Consider an arbitrary hamiltonian with finite-range interactions in an -dimensional lattice. The eigenstates have a well-defined global energy, but inside a region of the lattice the energy may fluctuate. (The nomenclature of FIG. 1 is followed.) In Section III it is proven that these fluctuations can be observed by measuring the exterior of the region and a superficial shell inside the region, that is . In Section IV a condition is imposed to the ground state: if the operator has support on the region which is separated from the support of the operator by a distance , then the connected correlation function decays at least as
where is a constant. This implies that energy fluctuations inside the region cannot be observed in its bulk, namely . This provides a characterization for the approximate support of the global ground state inside the region . In Section V a condition on the density of states is assumed: if is the subhamiltonian with all terms of whose support is fully contained in , then the number of eigenvalues lower than is bounded by
where is the lowest eigenvalue and some constants independent of . This condition is only assumed for . This implies an upper-bound on the dimension of the above-defined support subspace. This is used to bound the Von Newmann entropy for the reduction of the global ground state in the region
Section VI contains a simpler proof for the area law (3) without assuming (1), but assuming (2) for all the range of . In Section VII the above results for the ground state are generalized to other low-energy states (not necessarily eigenstates). Section VIII contains the conclusions.
Iii Locality and energy fluctuations
iii.1 Local interactions
Consider a system with one particle at each site of a finite -dimensional cubic lattice . The distance between two sites is defined by
In the case of periodic boundary conditions or hybrids, this distance has to be modified with the appropriate identification of sites. Each particle has associated a Hilbert space with finite dimension .
The hamiltonian of the system can be written as
where each term can have nontrivial support on first neighbors ( such that ). There is a constant which bounds the operator norm of all terms . (The operator norm of a matrix is equal to its largest singular value.) Translational symmetry is not assumed, so each term is arbitrary. The eigenstates and eigenvalues of are denoted by
where the index labels the eigenvalues in increasing oreder .
Note that any hamiltonian with finite-range interactions in a sufficiently regular lattice can be brought to the form of , by coarse-graining the lattice. Quantum field theories with local interactions can also be brought to the form of by lattice regularization. In the case of bosons, a truncation in the local degrees of freedom is needed. In the case of fermions, a multi-dimensional Jordan-Wigner transformation VC is needed.
iii.2 The Lieb-Robinson Bound
The hamiltonian satisfies the premises for the Lieb-Robinson Bound LR ; exp clustering . Let be two operators acting respectively on the regions , with . The distance between two regions is defined by
The time-evolution of an operator in the Heisenberg picture is . The Lieb-Robinson Bound states that
where . When the two operators almost commute. In other words, the dynamics generated by does not allow for the propagation of signals at speed much larger than . A simple proof of the Lieb-Robinson bound (8) is provided in Appendix C.
iii.3 Average for the energy fluctuations
For any region and any integer define the exterior, the boundary and the superficial shell as
respectively (see FIG. 1). The hamiltonian is defined as the sum of all terms whose support is fully contained in . The eigenstates and eigenvalues of are denoted by
where the index labels the eigenvalues in increasing order . The sum of all terms which simultaneously act on both, and , is denoted , and has norm . The expectation of any operator with the ground state is denoted by . Without loss of generality it can be assumed that each is positive semi-definite, which implies
This can be sumarized as follows.
The energy frustration of the global ground state in a region is, at most, proportional to the boundary .
iii.4 Observation of energy fluctuations
For any value of define the operator
where . The action of onto the global ground state implements an approximate projection onto the subspace with energy lower than inside the region ,
This integral is the error function, which is a soft step function. In the limit where the softness parameter tends to zero, the operator inside the square brackets becomes a projector. The operator has non-trivial support on the whole lattice , but remarkably, it can be approximated by the operator
which has non-trivial support only in the region . More quantitatively, the bound
is proven in Lemma 1 (Appendix), using techniques similar to the ones in 1D ; exp clustering ; spectrum cond . The fact that is solely a consequence of the locality of interactions and can be understood as follows. According to the Lieb-Robinson bound (8), if , any operator with support on evolves to an operator with approximate support on . Then , or in other words, the unitary in (14) approximately acts like the identity inside , or in other words , which justifies the definition (16).
The right-hand side of (16) is an average of unitaries, therefore . Then, the operators and define a two-outcome generalized measurement on , which tells whether the energy inside is below or above , approximately.
Everything shown in this section for the ground state generalizes to all eigenstates. The action of onto is
where . Summarizing, for each eigenstate there is an operator which approximately projects onto the subspace with energy inside the region , by only acting on the exterior and the shell . The degree of approximation increases with , the width of . The larger is, the closer and are, and the smaller the softness parameter is.
The energy fluctuations of an eigenstate inside a region can be observed by measuring the exterior and a superficial shell inside the region, that is (see FIG. 1).
Iv Support of the ground state inside a region
iv.1 Decay of correlations
It is usually the case that, when the system is in the ground state, the correlation between two observables acting on different sites decrease with the distance between the sites. Let be a function which upper-bounds the connected correlation function of any pair of operators acting respectively on the disjoint regions , with and ,
For the argument of this paper, both, the decay with the distance and the scaling with size of the support of the operators , are relevant. It is shown in exp clustering that any hamiltonian with an energy gap above the ground state has
with correlation length . To prove the area law for the entropy the following condition is needed.
Assumption 1 The correlation functions for the ground state decay at least as
where and are constants.
Note that both, (20) and (21), have the same relative scaling of and , but assumption (21) is weaker than (20). Although the decay (21) is polynomial in , it is not the correlation function of a critical hamiltonian, where one expects . Unfortunately, the argument of this paper does not give an area law with such scaling in .
iv.2 Energy fluctuations inside a region cannot be observed in its bulk
For any region and any integer define the extended region as
Suppose the existence of an operator with support in such that
This operator acts onto the ground state in a similar way as does, then the two operators are correlated
and their corresponding supports are separated by a distance . For the right choice of and large enough the existence of is in contradiction with Assumption 1, therefore
The energy fluctuations of the global ground state inside a region cannot be observed in the bulk of the region, that is .
In the following subsection, a quantitative example of this fact is given.
iv.3 Characterization of the support
Definition of For each eigenstate of with consider the Schmidt decomposition S ent with respect to the partition and . Define as the projector onto the subspace of generated by all vectors defined above, symbolically
Let be the projector onto the complementary subspace. Lemma 3 (Appendix) shows that the assignation (23) implies
Recalling that the respective supports of and are separated by a distance , one can invoke the decay of correlations (19) without specifying the function ,
for sufficiently large , where holds. Concluding, the support of the global ground state inside is contained in the subspace characterized by , up to some small weight (28).
iv.4 A renormalization group scheme
The projector defined above allows for certifiably-generating a low-energy effective theory for : the hamiltonian terms inside can be renormalized as
The whole lattice can be divided in similar regions, and the transformation (29) performed in each of them. The fidelity between the effective and the original ground-states can be bounded with (28), and increased by enlarging . As explained in Section VI, one can also obtain arbitrarily-good fidelities for any low-energy state.
V Entanglement in the ground state
v.1 Energy spectrum
In the previous section, a subspace which approximately contains the support of the ground state inside a region has been characterized. In order to bound its dimension, an additional assumption is needed: if the boundary conditions of the hamiltonian are left open, the number of eigenstates with vanishing energy-density must not be exponential in the volume.
Assumption 2 There are constants such that, for any region and energy
the number of eigenvalues of lower than satisfies
The area law is nontrivial when applied to regions such that , or equivalently . In this case, the eigenstates with energy proportional to the boundary (30) have vanishing energy density . According to spectrum cond , Assumption 2 holds for many systems that have an energy gap above the ground state. There are known hamiltonians which violate Assumption 2 and have a gap, but when the boundary conditions are opened there appears a degeneracy for the ground state which is exponential in the volume spectrum cond . Massive free bosons and fermions satisfy Assumption 2. Contrary, massless free frermions violate it as .
The factor in (31) can be understood with the following example. Consider the hamiltonian
where the subindex specifies in which site the matrix acts. The energy counts the number of local excitations, hence the degeneracy is the binomial of over , which can be upper-bounded by . The constant factor in (31) is introduced because some hamiltonians with open boundary conditions have a degeneracy (or approximate degeneracy) which is exponential in the size of the boundary.
Consider again the Schmidt decomposition of each eigenstate with respect to the partition and (Definition of ). The dimension of the Hilbert space is , therefore the support of each on has at most dimension . This and Assumption 2 provide a bound for the rank of the projector
v.2 Entropy of an arbitrary region
Consider a region being a completely arbitrary subset of the lattice. It not need to be convex, full-dimensional nor connected. For any site
Let be the reduction of the ground state in , and its eigenvalues in decreasing order. Assumptions 1 and 2 imply (21), (28) and (32), which impose the following constraints on the eigenvalues: for any integer ,
Now one can find the probability distribution which maximizes the entropy given the above constraints. This is done in Appendix B with the following result.
Result 1 The entropy of the reduction of the ground state inside an arbitrary region satisfies
v.3 Entropy of a cubic region
Consider the case where the chosen region is a hypercube . One can proceed as before, but the bounds analogous to (33) are smaller, implying a smaller bound for the entropy. All this is worked out in Appendix B.
Result 2 The entropy of the reduction of the ground state inside an cubic region satisfies
It is expected that the entropy of any full-dimensional convex region obeys the same scaling (36).
Vi Simpler proof for the area law
An area law can be easily proven without Assumption 1, if Assumption 2 is extended to all values of the energy , not only the ones satisfying (30). Let be the region where the entropy is estimated, and the sum of all terms of the hamiltonian (5) which are fully contained in . Following the conventions of this paper, the eigenstates and eigenvalues are denoted by , where . The strong version of Assumption 2 tells that all the eigenvalues satisfy
The global ground state can be written as
where the coefficients are non-negative and add up to one. It is shown in S ent that the entanglement entropy of is upper-bounded by the entropy of the -coefficients
Locality implies equation (13), which can be written as
the area law. This calculation is made in Appendix D.
Vii Entanglement in excited states
The entanglement properties of excited states have also been studied. In references DS1 ; DS2 ; DS3 the motivation was to study the robustness of the area law for the entropy of black holes. They show that in systems of free bosons, the entropy of some low-energy excited states scales at most like the area. In AFC the entropy scaling in one-dimensional spin systems is analyzed. They show that some excited states have entropy proportional to the volume, but low-energy states obey an area law. All this work is for integrable systems. In what follows, we address the general case.
Sometimes, low-lying excited states have correlation functions similar to the ones of the ground state. The single-mode ansatz for excitations with momentum is
where is an operator acting on site such that . If have support on finite regions and the volume of the system tends to infinite, then the correlation function (19) for the state (42) is the same as for . The same happens to excited states containing a small number of single-mode excitations. Examples of single-mode excitations are: spin waves, free bosons and free fermions. In this section it is shown that such excited states obey an area law similar to the one for the ground state. Actually, this is done with a bit more generality.
Consider an arbitrary superposition of eigenstates with bounded energy
Applying Assumption 1 to the state (43), the arguments follow exactly as for the ground state. Repeating the calculation of the entropy for a cubic region , and keeping track of the term proportional to one obtains
It is shown that ground states and low-energy states obey an area law for the entropy, provided two conditions hold: (i) the state has a sufficient decay of correlations, and (ii) the number of eigenstates with vanishing energy-density is not exponential in the volume of the system.
A universal property for local hamiltonians is also here established. The energy fluctuations of eigenstates inside an arbitrary region can be observed by measuring the exterior and a superficial shell of the region. This extends to any pure state that can be written as a superposition of eigenstates with similar energy.
Some thermodynamic quantities at finite temperature only depend on the density of states. Examples are: free energy, (global) entropy, heat capacity, etc. This paper establishes a relation between these thermodynamic quantities and ground-state entanglement.
Acknowledgements.The author is very thankful to Ignacio Cirac and Guifre Vidal. This work has been financially supported by Caixa Manresa. Additional support has come from the Spanish MEC project TOQATA (FIS2008-00784) and QOIT (Consolider Ingenio 2010), ESF/MEC project FERMIX (FIS2007-29996-E), EU Integrated Project SCALA, EU STREP project NAMEQUAM, ERC Advanced Grant QUAGATUA.
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Appendix A proofs
Proof First, express and with a single integral, by using the identity
Second, define the operators
which respectively act on the regions . Note that , , , and
This, the triangular inequality, Lemma 2, and the Lieb-Robinson bound (8), give
Putting and using Stirling’s approximation
Putting one obtains (46).
Lemma 2. Let be hermitian matrices and , then
Proof If is a differentiable function with then . This implies the following two equalities. The following two inequalities are a consequence of the triangular inequality for the operator norm.
Proof . The positive operator
allows for writing equality (15) as
The two projectors
with , satisfy
where we have used that . The positivity of and the second inequality in (53) imply
A worst-case estimation gives
Appendix B calculation of the entropy
b.1 Entropy of an arbitrary region
Consider the probability distribution defined by
for every integer , and
This distribution is uniform in blocks of the maximum size that constraints (34) allow. Then, it is the distribution satisfying (34) with maximum entropy. The upper-bound on the entropy of gets simplified by using the substitutions and
Using this, one obtains
b.2 Entropy of a cubic region
Appendix C The Lieb-Robinson bound
Let be two operators with support on the regions respectively, and . Let be an arbitrary operator and , where and is the hamiltonian (5). Using the Jacobi identity twice one obtains