Entanglement and ground states of gapped Hamiltonians
Spyridon Michalakis
备注：
91 pages, Ph.D. dissertation, University of California, Davis. Chapters 2 and 4 are extended versions of arXiv:math-ph/0606018v3 and arXiv:0707.1722v1 [quant-ph], respectively
We begin by deriving bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure Finitely Correlated States (FCS). The bounds we derive become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our result permits a more efficient calculation, numerically and in some cases even analytically, of the entanglement in arbitrary finitely correlated quantum spin chains. We continue the study of entanglement in the setting of ground states of Hamiltonians with a spectral gap. In particular, for $V$ a finite subset of $Z^d$, we let $H_V$ denote a Hamiltonian on $V$ with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap. For a density matrix $ρ_A$ that describes the finite-volume restriction to a region $A$ of the unique ground state, we provide a detailed version of a proof by M. Hastings, that the entropy of $ρ_A$ is bounded by a uniform constant $C$. Moreover, we provide a detailed generalization of the 1-dimensional construction of Hastings' approximation to the ground state in dimensions 2 and higher. Finally, we turn our attention to the study of a conjecture central to Quantum Information Theory, the multiplicativity of the maximal output Schatten $p$-norm of quantum channels, for $p > 1$. In particular, we study the output 2-norm for a special class of quantum channels, the depolarized Werner-Holevo channels, and show that multiplicativity holds for a product of two identical channels in this class.
We define a multiparty entanglement measure, called generalized geometric measure, that can detect and quantify genuine multiparty entanglement for any number of parties. The quantum phase transitions in exactly solvable models like the anisotropic XY model can be detected by this measure. We find that the multisite measure can be a useful tool to detect quantum phenomena in more complex systems like quasi 2D and 2D frustrated Heisenberg antiferromagnets. We propose an order parameter, called bound generalized geometric measure, in the spirit of bound quantum states, that can recognize the gap…
We give an analytical description of long distance entanglement (LDE) mediated by one-dimensional quantum spin chains recently found in numerical studies. We develop a formalism that allows the computation of LDE for weakly interacting probes with gapped many-body systems. At zero temperature, a DC response function determines the ability of the physical system to generate genuine quantum correlations between the probes. We show that the biquadratic Heisenberg spin-1 chain is able to produce LDE in the thermodynamical limit and that the finite antiferromagnetic Heisenberg chain maximally entan…
We consider general locally-interacting arbitrary-dimensional lattice spin systems that are gapped for any system size. We show under reasonable conditions that nondegenerate ground states of such systems obey the entanglement area law. In so doing, we offer an intuitive picture on how a spectral gap restricts the correlations that a ground state can accommodate and leads to such a special feature.
We prove the existence of gapped quantum Hamiltonians whose ground states exhibit an infinite entanglement length, as opposed to their finite correlation length. Using the concept of entanglement swapping, the localizable entanglement is calculated exactly for valence bond and finitely correlated states, and the existence of the so--called string-order parameter is discussed. We also report on evidence that the ground state of an antiferromagnetic chain can be used as a perfect quantum channel if local measurements on the individual spins can be implemented.
The single-copy entanglement of a given many-body system is defined [J. Eisert and M. Cramer, Phys. Rev. A. 72, 042112 (2005)] as the maximal entanglement deterministically distillable from a bipartition of a single specimen of that system. For critical (gapless) spin chains, it was recently shown that this is exactly half the von Neumann entropy [R. Orus, J. I. Latorre, J. Eisert, and M. Cramer, Phys. Rev. A 73, 060303(R) (2006)], itself defined as the entanglement distillable in the asymptotic limit: i.e. given an infinite number of copies of the system. It is an open question as to what the…
Topological order, reflected in long range patterns of entanglement, is quantified by the topological entanglement entropy (TEE) $γ$. We show that for gapped quantum spin liquids (QSL) it is possible to extract $γ$ using two-spin local correlators. We demonstrate our method for the gapped $\mathbb{Z}_2$ Kitaev spin liquid on a honeycomb lattice with anisotropic interactions. We show that the $γ= \log 2$ for $\mathbb{Z}_2$ topological order can be simply extracted from local two-spin correlators across two different bonds, with an accuracy comparable or higher than the Kitaev-Preskill construct…
Many-body entangled quantum states studied in condensed matter physics can be primary resources for quantum information, allowing any quantum computation to be realized using measurements alone, on the state. Such a universal state would be remarkably valuable, if only it were thermodynamically stable and experimentally accessible, by virtue of being the unique ground state of a physically reasonable Hamiltonian made of two-body, nearest neighbor interactions. We introduce such a state, composed of six-state particles on a hexagonal lattice, and describe a general method for analyzing its prop…
Entanglement between the constituents of a quantum system is an essential resource in the implementation of many quantum processes and algorithms. Indeed, universal quantum computation is possible by measuring individual qubits comprising highly entangled cluster states. In this work it is shown that the unique gapped ground state of non-interacting fermions hopping on a specially prepared lattice is equivalent to a cluster state, where the entanglement between qubits results solely by fermionic indistinguishability and antisymmetry. A deterministic strategy for universal measurement-based qua…
We investigate the scaling of the Rényi $α$-entropies in one-dimensional gapped quantum spin models. We show that the block entropies with $α> 2$ violate the area law monotonicity and exhibit damped oscillations. Depending on the existence of a factorized ground state, the oscillatory behavior occurs either below factorization or it extends indefinitely. The anomalous scaling corresponds to an entanglement-driven order that is independent of ground-state degeneracy and is revealed by a nonlocal order parameter defined as the sum of the single-copy entanglement over all blocks.
We construct a Hamiltonian whose ground state encodes a time-independent emulation of quan- tum teleportation. We calculate properties of the Hamiltonian, using exact diagonalization and a mean-field theory, and argue that it has a gap. The system exhibits an illuminating relationship to the well-known AKLT (Affleck, Lieb, Kennedy and Tasaki) model.
We introduce a family of Hamiltonian systems for measurement-based quantum computation with continuous variables. The Hamiltonians (i) are quadratic, and therefore two body, (ii) are of short range, (iii) are frustration-free, and (iv) possess a constant energy gap proportional to the squared inverse of the squeezing. Their ground states are the celebrated Gaussian graph states, which are universal resources for quantum computation in the limit of infinite squeezing. These Hamiltonians constitute the basic ingredient for the adiabatic preparation of graph states and thus open new venues for th…
The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable models for a plethora of gapped quantum phases of matter. Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weak local perturbations thus allows us to develop a general framework within which the entanglement features of gapped topological phases can be computed and contrasted. In particular, we study models harboring fracton orde…
We develop a theory of gapped domain wall between topologically ordered systems in two spatial dimensions. We find a new type of superselection sector -- referred to as the parton sector -- that subdivides the known superselection sectors localized on gapped domain walls. Moreover, we introduce and study the properties of composite superselection sectors that are made out of the parton sectors. We explain a systematic method to define these sectors, their fusion spaces, and their fusion rules, by deriving nontrivial identities relating their quantum dimensions and fusion multiplicities. We pro…
Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. We give the first randomized polynomial-time algorithm for finding ground states of gapped one-dimensional Hamiltonians: it outputs an (inverse-polynomial) approximation, expressed as a matrix product state (MPS) of polynomial bond dimension. The algorithm combines many ingredients, including recently discovered structural features of gapped 1D systems, convex programming, insights from classical algorithms for 1D satisfiability, and new techniques for manipulating and bounding the complexity of…
We investigate the behavior of genuine multiparticle entanglement, as quantified by the generalized geometric measure, in gapless-to-gapped quantum transitions of one- and two-dimensional quantum spin models. The investigations are performed in the exactly solvable one-dimensional $XY$ models, as well as two-dimensional frustrated $J_{1}-J_{2}$ models, including the Shastry-Sutherland model. The generalized geometric measure shows non-monotonic features near such transitions in the frustrated quantum systems. We also compare the features of the generalized geometric measure near the quantum cr…
A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length $N$ of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-$k$ segments of the chain, the reduced density matrices of our approximations are $ε$-close to thos…
Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In a previous work, we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-p…
We study the symmetry resolved entanglement entropies in gapped integrable lattice models. We use the corner transfer matrix to investigate two prototypical gapped systems with a U(1) symmetry: the complex harmonic chain and the XXZ spin-chain. While the former is a free bosonic system, the latter is genuinely interacting. We focus on a subsystem being half of an infinitely long chain. In both models, we obtain exact expressions for the charged moments and for the symmetry resolved entropies. While for the spin chain we found exact equipartition of entanglement (i.e. all the symmetry resolved …
A (deterministic) polynomial-time algorithm is proposed for approximating the ground state of (general) one-dimensional gapped Hamiltonians. Let $ε,n,η$ be the energy gap, the system size, and the desired precision, respectively. Neglecting $ε$-dependent subpolynomial (in $n$) and constant factors, the running time of the algorithm is $n^{O(1)}$ for $η=n^{-O(1)}$.
In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entan…
We investigate the entanglement properties of the valence-bond-solid states with generic integer-spin $S$. Using the Schwinger boson representation of the valence-bond-solid states, the entanglement entropy, the von Neumann entropy of a subsystem, is obtained exactly and its relationship with the usual correlation function is clarified. The saturation value of the entanglement entropy, $2 \log_2 (S+1)$, is derived explicitly and is interpreted in terms of the edge-state picture. The validity of our analytical results and the edge-state picture is numerically confirmed. We also propose a novel …
We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an `entropy density' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size, and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size independent constant con…
In this paper, we consider classification problem of asymmetric gapped Hamiltonians, which are given as the non-degenerate part of the Hamiltonians introduced in [O1]. We consider the $C^1$-classification, which takes into account the effect of boundaries. We show that the left and right degeneracies of edge ground states are the complete invariant. As a corollary, we consider the bulk-classification problem. We study Hamiltonians that1.are given by translation invariant finite range interactions, 2.are gapped in the bulk, 3.are frustration-free, 4.have uniformly bounded ground state degenerac…
We consider the $C^1$-classification of gapped Hamiltonians introduced in [Fannes-Nachtergaele-Werner, Nachtergaele] as parents Hamiltonians of translation invariant finitely correlated states. Within this family, we show that the number of edge modes, which is equal at the left and right edge, is the complete invariant. The construction proves that translation invariance of the `bulk' ground state does not need to be broken to establish $C^1$-equivalence, namely that the spin chain does not need to be blocked.
We extend our studies of holographic entropy inequalities to gapped phases of matter. For any number of regions, we determine the linear entropy inequalities satisfied by systems in which the entanglement entropy satisfies an exact area law. In particular, we find that all holographic entropy inequalities are valid in such systems. In gapped systems with topological order, the "cyclic inequalities" derived recently for the holographic entanglement entropy generalize the Kitaev-Preskill formula for the topological entanglement entropy. Finally, we propose a candidate linear inequality for gener…
We study the long-time evolution of the bipartite entanglement in translationally invariant gapped harmonic lattice systems with finite-range interactions. A lower bound for the von Neumann entropy is derived in terms of the purity of the reduced density matrix. It is shown that starting from an initially Gaussian state the entanglement entropy increases at least linearly in time. This implies that the dynamics of gapped (non-critical) harmonic lattice systems cannot be efficiently simulated by algorithms based on matrix-product decompositions of the quantum state.
We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.
In this paper the the effect of Dzyaloshinskii-Moriya interaction and anisotropy on the Entanglement of Heisenberg chain has been studied. While the anisotropy suppress the entanglement due to favoring of the alignment of spins, the DM interaction restores the spoiled entanglement via creation of the quantum fluctuations. Thermodynamic limit of the model and emerging of non-analytic behavior of the entanglement have also been probed. The singularities of the entanglement correspond to the critical boundary separating different phases of the model. The singularity of the entanglement derivative…
We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model $H_A$ in three dimensions, in order to understand long-range entanglement structure. The cubic code model has degenerate and locally indistinguishable ground states under periodic boundary conditions. In the entanglement renormalization, one applies local unitary transformations on a state, called disentangling transformations, after which some of the spins are completely disentangled from the rest and then discarded. We find a disentangling unitary to establish equivalenc…
We determine the entanglement capability of self-inverse Hamiltonian evolution, which reduces to the known result for Ising Hamiltonian, and identify optimal input states for yielding the maximal entanglement rate. We introduce the concept of the operator entanglement rate, and find that the maximal operator entanglement rate gives a lower bound on the entanglement capability of a general Hamiltonian.
We exploit the knowledge of the entanglement spectrum in the ground state of the gapped XXZ spin chain to derive asymptotic exact results for the full counting statistics of the transverse magnetisation in a spin block of length. We found that for a subsystem of even length the full counting statistics is Gaussian, while for odd subsystems it is the sum of two Gaussian distributions. We test our analytic predictions with accurate tensor networks simulations. As a byproduct, we also obtain the symmetry (magnetisation) resolved entanglement entropies.
We propose a scheme for a ground-code measurement-based quantum computer, which enjoys two major advantages. First, every logical qubit is encoded in the gapped degenerate ground subspace of a spin-1 chain with nearest-neighbor two-body interactions, so that it equips built-in robustness against noise. Second, computation is processed by single-spin measurements along multiple chains dynamically coupled on demand, so as to keep teleporting only logical information into a gap-protected ground state of the residual chains after the interactions with spins to be measured are turned off. We descri…
All Hamiltonian complexity results to date have been proven by constructing a local Hamiltonian whose ground state -- or at least some low-energy state -- is a "computational history state", encoding a quantum computation as a superposition over the history of the computation. We prove that all history-state Hamiltonians must be critical. More precisely, for any circuit-to-Hamiltonian mapping that maps quantum circuits to local Hamiltonians with low-energy history states, there is an increasing sequence of circuits that maps to a growing sequence of Hamiltonians with spectral gap closing at le…
We study the time evolution of the amount of entanglement generated by one dimensional spin-1/2 Ising-type Hamiltonians composed of many-body interactions. We investigate sets of states randomly selected during the time evolution generated by several types of time-independent Hamiltonians by analyzing the distributions of the amount of entanglement of the sets. We compare such entanglement distributions with that of typical entanglement, entanglement of a set of states randomly selected from a Hilbert space with respect to the unitarily invariant measure. We show that the entanglement distribu…
We quantify the capability of creating entanglement for a general physical interaction acting on two qubits. We give a procedure for optimizing the generation of entanglement. We also show that a Hamiltonian can create more entanglement if one uses auxiliary systems.
Recent results on the stability of the spectral gap under general perturbations for frustration-free Hamiltonians, have motivated the following question: Does the entanglement entropy of quantum states that are connected to states satisfying an area law along gapped Hamiltonian paths, also satisfy an area law? We answer this question in the affirmative, combining recent advances in quasi-adiabatic evolution and Lieb-Robinson bounds with ideas from the proof of the 1D area law.
We investigate a recent conjecture connecting the AdS/CFT correspondence and entanglement renormalization tensor network states (MERA). The proposal interprets the tensor connectivity of the MERA states associated to quantum many body systems at criticality, in terms of a dual holographic geometry which accounts for the qualitative aspects of the entanglement and the correlations in these systems. In this work, some generic features of the entanglement entropy and the two point functions in the ground state of one dimensional gapped systems are considered through a tensor network state. The te…
We construct a topological spin liquid (TSL) model on the kagome lattice, with SU(3) symmetry with the fundamental representation at each lattice site, based on Projected Entangled Pair States (PEPS). Using the PEPS framework, we can adiabatically connect the model to a fixed point model (analogous to the dimer model for Resonating Valence Bond states) which we prove to be locally equivalent to a $Z_3$ quantum double model. Numerical study of the interpolation reveals no sign of a phase transition or long-range order, characterizing the model conclusively as a gapped TSL. We further study the …
Numerical studies of the reduced density matrix of a gapped spin-1/2 Heisenberg antiferromagnet on a two-leg ladder find that it has the same form as the Gibbs density matrix of a gapless spin-1/2 Heisenberg antiferromagnetic chain at a finite temperature determined by the spin gap of the ladder. We investigate this interesting result by considering a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density mat…
The notion of capacity of entanglement is the quantum information theoretic counterpart of the heat capacity which is defined as the second cumulant of the entanglement spectrum. Given any bipartite pure state, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the reduced state of any of the subsystems. Here, we study the dynamics of this quantity under non-local Hamiltonian. Specifically, we address the question: Given an arbitrary non-local Hamiltonian what is the capacity of entanglement that the system can possess? As an useful application, we show th…
A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) -- the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science. The Quantum Lovász Local Lemma (QLLL) provides a sufficient condition for …
We provide the first solution of a time-dependent metric operator for the non-Hermitian Jaynes-Cummings Hamiltonian. We use this solution to calculate the entanglement between two identical isolated such Hamiltonians. The presence of a non-Hermitian interaction term leads to a spontaneously broken $\mathcal{PT}$-symmetric regime which manifests itself in the exotic time-evolution of entanglement. When the symmetry is broken, oscillatory modes transition into decay. As such that there is a drastic difference in behaviour between the broken and unbroken regimes.
It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit "symmetry fractionalization". Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices---which act on some virtual degrees of freedom---can transform under a projective representation. This was instrumental in classifying gapped symmetry protected phases that manifest in one dimensional quantum many-body systems. Here we consider the multi-scale entanglement renormalization ansatz (MERA) description of …
This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protec…
We analyse in detail the effect of non-trivial band topology on the area law behaviour of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model into 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semi-metallic p…
Applying a symmetric bulk bipartition to one-dimensional Affleck-Kennedy-Lieb-Tasaki valence bond solid (VBS) states for the integer spin-S Haldane gapped phase, we can create an array of fractionalized spin-S/2 edge states with the super unit cell $l$ in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with $l=even$ corresponds to the quantum antiferromagnetic Heisenberg spin-S/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the …
We introduce a phenomenological theory for many-body control of critical phenomena by engineering causally-induced gaps for quantum Hamiltonian systems. The core mechanisms are controlling information flow within and/or between clusters that are created near a quantum critical point. To this end, we construct inhomogeneous quantum phase transitions via designing spatio-temporal quantum fluctuations. We show how non-equilibrium evolution of disordered quantum systems can create new effective correlation length scales and effective dynamical critical exponents. In particular, we construct a clas…
We propose an efficient algorithm for the ground state of frustration-free one-dimensional gapped Hamiltonians. This algorithm is much simpler than the original one by Landau et al., and thus may be easily accessible to a general audience in the community. We present all the details in two pages.
