We study quantum entanglement in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model defined on two-dimensional graphs with reflection and/or inversion symmetry. The ground state of this spin model is known as the valence-bond-solid state. We investigate the properties of reduced density matrix of a subsystem which is a mirror image of the other one. Thanks to the reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is $\ln 2$. This deviation is interpreted in terms of the hidden spin chain along the boundary between the subsystems. In some cases where graphs are on ladders, the numerical results are analytically or algebraically confirmed.
We introduce for SU(2) quantum spin systems the Valence Bond Entanglement Entropy as a counting of valence bond spin singlets shared by two subsystems. For a large class of antiferromagnetic systems, it can be calculated in all dimensions with Quantum Monte Carlo simulations in the valence bond basis. We show numerically that this quantity displays all features of the von Neumann entanglement entropy for several one-dimensional systems. For two-dimensional Heisenberg models, we find a strict area law for a Valence Bond Solid state and multiplicative logarithmic corrections for the Neel phase.
We introduce a general method to construct one-dimensional translationally invariant valence bond solid states with a built-in Lie group $G$ and derive their matrix product representations. The general strategies to find their parent Hamiltonians are provided so that the valence bond solid states are their unique ground states. For quantum integer spin-$S$ chains, we discuss two topologically distinct classes of valence bond solid states: One consists of two virtual SU(2) spin-$J$ variables in each site and another is formed by using two $SO(2S+1)$ spinors. Among them, a new spin-1 fermionic v…
By relating the ground state of Temperley-Lieb hamiltonians to partition functions of 2D statistical mechanics systems on a half plane, and using a boundary Coulomb gas formalism, we obtain in closed form the valence bond entanglement entropy as well as the valence bond probability distribution in these ground states. We find in particular that for the XXX spin chain, the number N_c of valence bonds connecting a subsystem of size L to the outside goes, in the thermodynamic limit, as <N_c> = (4/pi^2) ln L, disproving a recent conjecture that this should be related with the von Neumann entropy, …
We study maximal bipartite entanglement in valence-bond states and show that the average bipartite entanglement $E_v^2$, between a sub-system of two spins and the rest of the system, can be maximized through a homogenized superposition of the valence-bond states. Our derived maximal $E_v^2$ rapidly increases with system size and saturates at its maximum allowed value. We also demonstrate that our maximal $E^2_v$ states are ground states of an infinite range Heisenberg model (IRHM) and represent a new class of resonating-valence-bond (RVB) states. The entangled RVB states produced from our IRHM…
A simple and efficient method for calculating the ground state for a class of antiferromagnet systems is presented. It combines the valence bond structure of the ground state for this class of systems and real space renormalization group. As an example, the entire Haldane phase for a spin-1 chain is described, from the AKLT model through to the Heisenberg model, ending at the critical WZNW $SU_2(2)$ model. The picture that emerges from this new method leads us to consider the relationship between the hidden topological order that characterizes the Haldane phase and potential applications in qu…
The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely related to quantum entanglement. We investigate measures of entanglement entropy based on transition graphs, which characterize state overlaps in the overcomplete valence-bond basis. The transition graphs can be generated using projector Monte Carlo simulations of ground states of specific hamiltonians or using importance-sampling of valence-bond configurations of amplitude-product states. We consider definitions of entanglement entropy based on the bonds or loops shared by two subsystems (bipartite entanglement). Re…
We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence-bond states proposed in the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest neighbor interactions, and the partially dimerized or AKLT/VBS (Valence Bond Solid) states,…
In this paper we provide the analytical derivation of the global geometric entanglement per block for the valence bond solid ground state of the spin-1 AKLT chain. In particular, we show that this quantity saturates exponentially fast to a constant when the sizes of the blocks are sufficiently large. Our result provides the first known example of an analytical calculation of the geometric entanglement for a gapped quantum many-body system in one dimension and far away from a quantum critical point.
We study the competition between antiferromagnetic order and valence bond solid formation in a two-dimensional frustrated spin-1/2 model. The J1-J2 model on the square lattice is further frustrated by introducing products of three-spin projectors which stabilize four dimer-product states as degenerate ground states. These four states are reminiscent of the dimerized singlet ground state of the Shastry-Sutherland model. Using exact diagonalizations, we study the evolution of the ground state by varying theratio of interactions. For a large range of parameters (J2 > 0.25J1), our model shows a di…
We present a theory of a hybrid quantum liquid state, $\textit{quantum spin-dipole liquid}$ (QSDL), in a hydrogen-bonded electron system, by combining a quantum proton ice and Anderson's resonating valence bond spin liquid theory, motivated by the recent experimental discovery of a quantum spin liquid with proton fluctuations in $κ$-H$_3$(Cat-EDT-TTF)$_2$ (a.k.a. H-Cat). In our theory, an electron spin liquid and a proton dipole liquid are realized simultaneously in the ground state called $\textit{quantum valence bond ice}$. In this state, neither of them can be established independently of t…
We investigate the entanglement properties of resonating-valence-bond states on two and higher dimensional lattices, which play a significant role in our understanding of various many-body systems. We show that these states are genuinely multipartite entangled, while there is only a negligible amount of two-site entanglement. We comment on possible physical implications of our findings.
We present a direct comparison of the recently-proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin 1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy, hence it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entangle…
Topological qubits based on $SU(N)$-symmetric valence-bond solid models are constructed. A logical topological qubit is the ground subspace with two-fold degeneracy, which is due to the spontaneous breaking of a global parity symmetry. A logical $Z$-rotation by angle $\frac{2π}{N}$, for any integer $N > 2$, is provided by a global twist operation, which is of topological nature and protected by the energy gap. A general concatenation scheme with standard quantum error-correction codes is also proposed, which can lead to better codes. Generic error-correction properties of symmetry-protected to…
We propose a simple one-dimensional spin-2 Hamiltonian, which exhibits two topologically distinct valence bond solid states in different exactly solvable limits. We then construct the phase diagram and study the quantum phase transition between these states using the infinite time evolving block decimation algorithms. From the scaling relation between the entanglement entropy and correlation length, we find that the central charge for the underlying critical conformal field theory is $c=2$.
The nature of the kagome Heisenberg antiferromagnet (HAFM) is under ongoing debate. While recent evidence points towards a Z_2 topological spin liquid, the exact nature of the topological phase is still unclear. In this paper, we introduce semionic Resonating Valence Bond (RVB) states, this is, Resonating Valence Bond states which are in the Z_2 ordered double-semion phase, and study them using Projected Entangled Pair States (PEPS). We investigate their physics and study their suitability as an ansatz for the HAFM, as compared to a conventional RVB state which is in the Toric Code Z_2 topolog…
Here we evaluate the many-body entanglement properties of a generalized SU(n) valence bond solid state on a chain. Our results follow from a derivation of the transfer matrix of the system which, in combination with symmetry properties, allows for a new, elegant and straightforward evaluation of different entanglement measures. In particular, the geometric entanglement per block, correlation length, von Neumann and Rényi entropies of a block, localizable entanglement and entanglement length are obtained in a very simple way. All our results are in agreement with previous derivations for the SU…
We extend the definition of the recently introduced valence bond entanglement entropy to arbitrary SU(2) wave functions of S=1/2 spin systems. Thanks to a reformulation of this entanglement measure in terms of a projection, we are able to compute it with various numerical techniques for frustrated spin models. We provide extensive numerical data for the one-dimensional J1-J2 spin chain where we are able to locate the quantum phase transition by using the scaling of this entropy with the block size. We also systematically compare with the scaling of the von Neumann entanglement entropy. We fina…
We introduce a hybrid quantum-classical variational algorithm to simulate ground-state phase diagrams of frustrated quantum spin models in the thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where the wave function of the cluster is provided by a parameterized quantum circuit whose key ingredient is a two-qubit real XY gate allowing to efficiently generate valence-bonds on nearest-neighbor qubits. Additional tunable single-qubit Z- and two-qubit ZZ-rotation gates allow the description of magnetically ordered and paramagnetic phases while restricting the variational opti…
In order to quantify entanglement between two parts of a quantum system, one of the most used estimator is the Von Neumann entropy. Unfortunately, computing this quantity for large interacting quantum spin systems remains an open issue. Faced with this difficulty, other estimators have been proposed to measure entanglement efficiently, mostly by using simulations in the valence-bond basis. We review the different proposals and try to clarify the connections between their geometric definitions and proper observables. We illustrate this analysis with new results of entanglement properties of spi…
We investigate the behavior of bipartite as well as genuine multipartite entanglement of a resonating valence bond state on a ladder. We show that the system possesses significant amounts of bipartite entanglement in the steps of the ladder while no substantial bipartite entanglement is present in the rails. Genuine multipartite entanglement present in the system is negligible. The results are in stark contrast with the entanglement properties of the same state on isotropic lattices in two and higher dimensions, indicating that the geometry of the lattice can have important implications on the…
We study controlled generation and measurement of superfluid d-wave resonating valence bond (RVB) states of fermionic atoms in 2D optical lattices. Starting from loading spatial and spin patterns of atoms in optical superlattices as pure quantum states from a Fermi gas, we adiabatically transform this state to an RVB state by change of the lattice parameters. Results of exact time-dependent numerical studies for ladders systems are presented, suggesting generation of RVB states on timescale smaller than typical experimental decoherence times.
We discuss topological disentangler for S=1 quantum spin chains in the Haldane phase. We first point out that Kennedy-Tasaki's(KT) nonlocal unitary transformation is the perfect disentangler for Affleck-Kennedy-Lieb-Tasaki model. We then demonstrate that the KT transformation can be reconstructed as an assembly of pair disentanglers. Finally, we show that the KT transformation can be regarded as a topological disentangler, which selectively disentangles the double-fold degeneracy in the entanglement spectrum of the S=1 Heisenberg chain.
The spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) valence-bond state on the hexagonal lattice was shown to be a universal resource state for measurement-based quantum computation (MBQC). Can AKLT states of higher spin magnitude support universal MBQC? We demonstrate that several hybrid 2D AKLT states involving mixture of spin-2 and other lower-spin entities, such as spin-3/2 and spin-1, are also universal for MBQC. This significantly expands universal resource states in the AKLT family. Even though frustration may be a hinderance to quantum computational universality, lattices can be modified to…
We introduce valence bond fluctuations, or bipartite fluctuations associated to bond-bond correlation functions, to characterize quantum spin liquids and their entanglement properties. Using analytical and numerical approaches, we find an identical scaling law between valence bond fluctuations and entanglement entropy in the two-dimensional Kitaev spin model and in one-dimensional chain analogues. We also show how these valence bond fluctuations can locate quantum phase transitions between the three gapped and the gapless Majorana semi-metal phases in the honeycomb model. We then study the eff…
Cluster states are entangled multipartite states which enable to do universal quantum computation with local measurements only. We show that these states have a very simple interpretation in terms of valence bond solids, which allows to understand their entanglement properties in a transparent way. This allows to bridge the gap between the differences of the measurement-based proposals for quantum computing, and we will discuss several features and possible extensions.
We study entanglement in Valence-Bond-Solid state. It describes the ground state of Affleck, Kennedy, Lieb and Tasaki quantum spin chain. The AKLT model has a gap and open boundary conditions. We calculate an entropy of a subsystem (continuous block of spins). It quantifies the entanglement of this block with the rest of the ground state. We prove that the entanglement approaches a constant value exponentially fast as the size of the subsystem increases. Actually we proved that the density matrix of the continuous block of spins depends only on the length of the block, but not on the total siz…
We investigate magnetic properties of strongly interacting four component spin-3/2 ultracold fermionic atoms in the Mott insulator limit with one particle per site in an optical lattice with honeycomb symmetry. In this limit, atomic tunneling is virtual, and only the atomic spins can exchange. We find a competition between symmetry breaking and liquid like disordered phases. Particularly interesting are valence bond states with bond centered magnetizations, situated between the ferromagnetic and conventional valence bond phases. In the framework of a mean-field theory, we calculate the phase d…
The Valence-Bond-Solid (VBS) states are in general ground states for certain gapped models. We consider the entanglement of VBS states on a two-dimensional Cayley tree. We show that the entropy of the reduced density operator does not depend on the whole size of the Cayley tree. We also show that asymptotically, the entropy is liearly proportional to the number of singlet states cut by the reduced density operator of the VBS state.
The concept of valence bond resonance plays a fundamental role in the theory of the chemical bond and is believed to lie at the heart of many-body quantum physical phenomena. Here we show direct experimental evidence of a time-resolved valence bond quantum resonance with ultracold bosonic atoms in an optical lattice. By means of a superlattice structure we create a three-dimensional array of independent four-site plaquettes, which we can fully control and manipulate in parallel. Moreover, we show how small-scale plaquette resonating valence bond states with s- and d-wave symmetry can be create…
Supersymmetric valence bond solid models are extensions of the VBS model, a paradigmatic model of `solvable' gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin/charge excitation spectrum, superconducting order parameter. A generalized hidden order is proposed, and the corresponding string n…
The d-level or qudit one-way quantum computer (d1WQC) is described using the valence bond solid formalism and the generalised Pauli group. This formalism provides a transparent means of deriving measurement patterns for the implementation of quantum gates in the computational model. We introduce a new universal set of qudit gates and use it to give a constructive proof of the universality of d1WQC. We characterise the set of gates that can be performed in one parallel time step in this model.
A basic problem of quantum communication is to generate entangled states between distant sites. Using entanglement swapping, we are able to generate an entangled state of the desired distance from connecting many short distance entangled states. We investigate the entanglement swapping of the Valence-Bond Solid state with un-identical local filtering operations. It is found that not only a long distance entangled state can be generated from the VBS state, but also there is a trade-off between the probability and the degree of the entanglement in the resultant state. The results are also genera…
We investigate the entanglement properties of resonating valence bond states on a two dimensional lattice in the presence of dopants that remove electrons from the lattice creating "holes". The movement of the holes generated by the Hubbard Hamiltonian in the regime of strong Coloumb repulsion in this setting could be responsible for the phenomenon of high temperature superconductivity as hypothesised by Anderson in Science {\bf 235}, 1196, (1987). We argue that there is a particular density of dopants (holes) where the entanglement contained in the lattice attains its maximal value for the ne…
We investigate entanglement properties in the ground state of the open/periodic SU($n$) generalized valence-bond-solid state consisting of representations of SU($n$). We obtain exact expression for the reduced density matrix of a block of contiguous spins and explicitly evaluate the von Neumann and the Rényi entropies. We discover that the Rényi entropy is independent of the parameter $α$ in the limit of large block sizes and its value $2 \log n$ coincides with that of von Neumann entropy. We also find the direct relation between the reduced density matrix of the subsystem and edge states for …
We study Gaussian valence bond states of continuous variable systems, obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites, applied at each of $N$ sites of an harmonic chain. The entanglement distribution in Gaussian valence bond states can be controlled by varying the input amount of entanglement engineered in a (2M+1)-mode Gaussian state known as the building block, which is isomorphic to the projector applied at a given site. We show how this mechanism can be interpreted in terms of multiple entanglement swappi…
Using a graphical presentation of the spin $S$ one dimensional Valence Bond Solid (VBS) state, based on the representation theory of the $SU(2)$ Lie-algebra of spins, we compute the spectrum of a mixed state reduced density matrix. This mixed state of two blocks of spins $A$ and $B$ is obtained by tracing out the spins outside $A$ and $B$, in the pure VBS state density matrix. We find in particular that the negativity of the mixed state is non-zero only for adjacent subsystems. The method introduced here can be generalized to the computation of entanglement properties in Levin-Wen models, that…
A resonating valence bond (RVB) state of a lattice of quantum systems is a potential resource for quantum computing and communicating devices. It is a superposition of singlet, i.e., dimer, coverings - often restricted to nearest-neighbour ones - of the lattice. We develop a polynomial representation of multipartite quantum states to prove that RVB states on ladder lattices possess genuine multipartite entanglement. The multipartite entanglement of doped RVB states and RVB states that are superposed with varying weights for singlet coverings of ladder lattices can both be detected by using thi…
