Quantum Spin Systems from the point a view of Quantum Information Theory

Quantum Spin Systems from the point a view of Quantum Information Theory Frank Verstraete, Ignacio Cirac Max-Planck-Institut für Quantenoptik

Overview • Entanglement verus correlations in quantum spin systems: • Localizable entanglement • Diverging entanglement length for gapped quantum systems • Valence bond states / Projected entangled pair states (PEPS) • In Spin chains • In quantum information theory • Coarse-graining (RG) of PEPS • PEPS as variational ground states • Illustration: RG + DMRG • Extending DMRG to periodic boundary conditions, time-evolution, finite-T • Using quantum parallelism for simulating 1-D quantum spin glasses • Simulation of 2-D quantum spin systems • Conclusion

Motivation • Many interesting phenomena in condensed matter occur in regime with strong correlations (e.g. quantum phase transitions) • Hard to describe ground states due to exponentially large Hilbert space • Powerful tool: study of 2-point correlation functions (length scale) • Central object of study in Quantum Information Theory: entanglement or quantum correlations • It is a resource that is the essential ingredient for e.g. quantum cryptography and quantum computing • Quantifies quantum nonlocality • Can QIT shed new light on properties of strongly correlated states as occurring in condensed matter?

Entanglement versus correlations Consider the ground state of e.g. a 1-D quantum Heisenberg Hamiltonian • Natural question in Statistical Mechanics: what are the associated correlation functions? • correlation functions of the form play central role: related to thermodynamic properties, to cross sections, detect long-range order and quantum phase transitions, define length scale … • Natural question in QIT: what is the amount of entanglement between separated spins (qubits) in function of their distance?

Quantum Repeater Briegel, Dür, Cirac, Zoller ‘98 • Spin Hamiltonians could also effectively describe a set of e.g. coupled cavities used as a quantum repeater: • The operationally motivated measure is in this case: how much entanglement is there between the first atom and the last one?

Entanglement in spin systems • Simplest notion of entanglement would be to study mixed state entanglement between reduced density operators of 2 spins • Problem: does not reveal long-range effect (Osborne and Nielsen 02, Osterloch et al. 02) • Natural definition of entanglement in spin systems from the resource point a view: localizable entanglement (LE) • Consider a state , then the LE is variationally defined as the maximal amount of entanglement that can be created / localized, on average, between spins i and j by doing LOCAL measurements on the other spins • Entanglement length: quantifies the distance at which useful entanglement can be created/localized, and hence the quality of a spin chain if used as a quantum repeater/channel Verstraete, Popp, Cirac 04

Entanglement versus correlations • LE quantifies quantum correlations that can be localized between different spins; how is this related to the “classical” correlations studied in quantum statistical mechanics? • Theorem: the localizable entanglement is always larger than or equal to the connected 2-point correlation functions • Consequences: • Correlation length is a lower bound to the Entanglement length: long-range correlations imply long-range entanglement • Ent. Length is typically equal to Corr. Length for spin ½ systems • LE can detect new phase transitions when the entanglement length is diverging but correlation length remains finite • When constructing a quantum repeater between e.g. cavities, the effective Hamiltonian should be tuned to correspond to a critical spin chain Verstraete, Popp, Cirac 04

Singlet Proj. in sym. subspace Illustration: the spin-1 AKLT-model • All correlation functions decay exponentially • The symmetric subspace is spanned by 3 Bell states, and hence this ground state can be used as a perfect quantum repeater • Diverging entanglement length but finite correlation length • LE detects new kind of long range order • Antiferromagnetic spin chain is a perfect quantum channel Verstraete, Martin-Delgado, Cirac 04

Pi+1 Pi+2 Pi+3 Pi+4 Pi+5 Pi+6 Pi Generalizing the AKLT-state: PEPS • Every state can be represented as a Projected Entangled Pair State (PEPS) as long as D is large enough • Extension to mixed states: take Completely Positive Maps (CPM) instead of projectors : • 1-D PEPS reduce to class of finitely correlated states / matrix product states (MPS) in thermodynamic limit (N!1) when P1=P2=L =P1 • Systematic way of constructing translational invariant states • MPS become dense in space of all states when D!1 • yield a very good description of ground states of 1-D systems (DMRG) Ex.: 5 qubit state Fannes, Nachtergaele, Werner 92

PEPS in higher dimensions: Verstraete, Cirac 04

Basic properties of PEPS • Correlation functions for 1-D PEPS can easily be calculated by multiplying transfer matrices of dimension D2 : • Number of parameters grows linearly in number of particles (NdDc) with c coordination number of lattice • 2-point correlations decay exponentially • Area law: entropy of a block of spins is proportional to its surface

Localizable Entanglement of VBS • Optimal measurement basis in context of LE is determined by the basis that maximizes the entanglement of assistance of the operator This is indeed the measurement that will optimize the quality of entanglement swapping Proj. P in phys. subspace DiVincenzo, Fuchs, Mabuchi, Smolin, Thapliyal, Uhlmann 98 (LE with more common entanglement measures can be calculated using combined DMRG/Monte Carlo method )

VBS in QIT • VBS play a crucial role in QIT: all stabilizer/graph/cluster states are simple VBS with qubit bonds • Gives insight in their decoherence properties, entropy of blocks of spins ... • Examples GHZ 5-qubit ECC

Measurement/Teleportation based quantum computation • Implementing local unitary U: • Implementing phase gate: • As Pauli operators can be pulled through Uph , this proves that 2- and 3-qubit measurements on a distributed set of singlets allows for universal QC Gottesman and Chuang ’99; Verstraete and Cirac 03

Measurement based quantum computation • Can joint measurements be turned into local ones at the expense of initially preparing a highly entangled state? • Yes: interpret logical qubits and singlets as virtual qubits and bonds of a 2-D VBS • Local measurements on physical qubits correspond to Bell/GHZ-measurements on virtual ones needed to implement universal QC • This corresponds exactly to the cluster-state based 1-way computer of Raussendorf and Briegel, hence unifying the different proposals for measurement based QC Raussendorf and Briegel ’01; Verstraete and Cirac 03; Leung, Nielsen et al. 04

Spin systems: basic properties • Hilbert space scales exponentially in number of spins • Universal ground state properties: • Entropy of block of spins / surface of block (holographic principle) • Correlations of spins decay typically exponentially with distance (correlation length) • The N-particle states with these properties form a tiny subspace of the exponentially large Hilbert space

Ground states are extreme points of a convex set: • Problem of finding ground state energy of all nearest-neighbor transl. invariant Hamiltonians is equivalent to characterizing the convex set of n.n. density operators arising from transl. invariant states • Finitely Correlated States / Matrix Product States / Projected Entangled Pair States provide parameterization that only grows linearly with number of particles but captures these desired essential features very well The Hamiltonian defines a hyperplane in (2s+1)2 dim. space

Pi+1 Pi+2 Pi+3 Pi+4 Pi+5 Pi+6 Pi PEPS as variational trial states for ground states of spin systems • All expectation values and hence the energy are multi-quadratic in the variables Pk • Strategy for minimizing energy for N-spin state: • Fix all projectors Pi except the jth • Both the energy and the norm are quadratic functions of the variable Pj and hence the minimal energy by varying Pi can be obtained by a simple generalized eigenvalue problem: Heff and N are function of the Hamiltonian and all other projectors, and can efficiently be calculated by the transfer matrix method • Move on to the (j§1)th particle and repeat previous steps (sweep) until convergence Verstraete, Porras, Cirac 04

J P0 P1 J l1 P1 P2 P3 P4 P5 P6 P0 P1 P2 P0 J l1 l2 l3 l5 l6 l4 Illustration 1 L • Wilson’s Renormalization Group (RG) for Kondo-effect: • RG calculates effective Hamiltonian by projecting out high energy modes; the effective Hamiltonian is spanned by a set of PEPS • Very successful for impurity problems, demonstrating validity of PEPS-ansatz

Illustration 2: DMRG White 92 • Most accurate method for determining ground states properties of 1-D spin chains (e.g. Heisenberg chains, Hubbard, …) • PEPS-approach proves the variational nature of DMRG • Numerical effort to find ground state is related to the amount of entanglement in a block of spins (Osborne and Nielsen 02, Vidal et al. 03)

P1 P2 P3 P4 P5 L PN DMRG and periodic boundary conditions • DMRG with periodic instead of open boundary conditions: Exactly translational invariant states are obtained, which seems to be important for describing dynamics Computational cost: ND5 versus ND3 (OBC)

Pi+1 Pi+2 Pi+3 Pi+4 Pi+5 Pi+6 Pi Further extensions: • Variational way of Calculating Excitations and dynamical correlation functions / structure factors using PEPS • Variational time evolution algorithms: (see also Vidal et al.) • Basic trick: variational dimensional reduction of PEPS • Given a PEPS |yDi of dimension D, find the one |cD’i of dimension D’< D such that || |yDi-|cD’i ||2 is minimized • This can again be done efficiently in a similar iterative way, yielding a variational and hence optimal way of treating time-evolution within the class of PEPS

Variational Dimensional Reduction of PEPS • Given a PEPS |yDi parameterized by the D£D matrices Ai, find the one |cD’i parameterized by D’£D’matrices Bi (D’< D) such as to minimize • Fixing all Bi but one to be optimized, this leads to an optimization of the form xy Heffx-xy y , with solution: Heffx=y/2 ; iterating this leads to global optimum • The error of the truncation can exactly be calculated at each step! • In case of OBC: more efficient due to orthonormalization • In the case of OBC, the algorithms of Vidal, Daley et al., White et al. are suboptimal but a factor of 2-3 times faster; a detailed comparison should be made

Finite-T DMRG: imaginary/real time evolution of a PEPS-purification: • Ancilla’s can also be used to describe quantum spin-glasses: due to quantum parallelism, one simulation run allows to simulate an exponential amount of different realizations; the ancilla’s encode the randomness

Simulation of 2-D quantum systems • Standard DMRG approach: trial state of the form Problems with this approach: dimension of bonds must be exponentially large: - area theorem - only possibility to get large correlations between vertical nearest neighbors We propose trial PEPS states that have bonds between all nearest neighbors, such that the area theorem is fulfilled by construction and all neighbors are treated on equal footing

P12 P13 P14 P15 P11 P22 P24 P25 P21 P23 P31 P32 P34 P35 P33 P45 P31 P44 P41 P42 • The energy of such a state is still a multi-quadratic function of all variable, and hence the same iterative variational principle can be used • The big difference: the determination of Heff and N is not obtained by multiplying matrices, but contracting tensors • This can be done using the variational dimensional reduction discussed before; note that the error in the truncation is completely controlled No (sign) problem with frustrated systems! Possible to devise an infinite dimensional variant

Alternatively, the ground state can be found by imaginary time evolution on a pure 2-D PEPS • This can be implemented by Trotterization; the crucial ingredient is again the variational dimensional reduction; the computational cost scales linearly in the number of spins: D10 • The same algorithm can of course be used for real-time evolution and for finding thermal states. • Dynamical correlation functions can be calculated as in the 1-D PEPS case • We have done simulations with the Heisenberg antiferromagnetic interaction and a frustrated version of it on 4£4, 10£10 and 20£20 • We used bonds of dimension 2,3,4; the error seems to decay exponentially in D • Note that we get mean field if D=1 • The number of variational parameters scales as ND4 and we expect the same accuracy as 1-D DMRG with dimension of bonds D2

10x10 20x20 4x4 4x4: 36.623 10x10: 2.353 (D=2); 2.473 (D=3) 20x20: 2.440 (D=2); 2.560 (D=3)

Wilson’s RG on the level of states: Coarse-graining PEPS • Goal: coarse-graining of PEPS-ground states • This can be done exactly, and leads to a fixed point exponentially fast; the fixed points are scale-invariant. This procedure is equivalent to Wilson’s numerical RG procedure • The fixed point of the generic case consists of the virtual subsystems becoming real, and where the ME-states are replaced with states with some entropy determined by the eigenvectors of the transfer matrix; note that no correlations are present • A complete classification of fixed points in case of qubit bonds has been made; special cases correspond to GHZ, W, cluster and some other exotic states in QIT

Conclusion • PEPS give a simple parameterization of multiparticle entanglement in terms of bipartite entanglement and projectors • Examples of PEPS: Stabilizer, cluster, GHZ-states • QIT-approach allows to generalize numerical RG and DMRG methods to different settings, most notably to higher dimensions

Quantum Spin Systems from the point a view of Quantum Information Theory