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Quantum Information Theory and Strongly Correlated Quantum Systems Frank Verstraete University of Vienna Strongly Correlated Quantum Systems Quantum Mechanics Quantum Chemistry, Statistical Physics N-representability Microscopic behavior, quantum Spin systems, Macroscopic behavior, order Hubbard models correlations, superposition parameters, phase transitions, Topological quantum principle order Ground States Quantum Error Correction, quantum coherence in Renormalization Group macroscopic systems (NRG, DMRG) Frustration, monogamy of entanglement, Simulation Stochastic non-equilibrium cryptography systems Entanglement PEPS Area Laws Quantum Information Entropy Theory Complexity Theory Information Theory Shannon Entropy, channel capacity Entanglement Complementary viewpoints on entanglement: • Quantum information theory: it is a resource that allows for revolutionary information theoretic tasks • Quantum many-body physics: entanglement gives rise to exotic phases of matter • Numerical simulation of strongly correlated quantum systems: enemy nr. 1! Of course these viewpoints are mutually compatible: - Complexity of simulation vs. power of quantum computation - Topological quantum order vs. quantum error correction Key question: what kind of superpositions appear in nature? Hilbert space is a convenient illusion • Let’s investigate the features of the manifold of states that can be created under the evolution H(t) for times T polynomial in N: T= Nd • Solovay-Kitaev: given a standard universal gate set on N spins (cN gates), then any 2-body unitary can be approximated with log(1/ε) standard gates such that ║U-Uε║< ε • Given any quantum circuit acting on pairs and of polynomial depth Nd, this can be reproduced up to error ε by using Nd log(Nd /ε) standard gates. The total number of states that can hence be created using that many gates scales as Nd cN N d log • Consider however the DN dimensional hypersphere; the number of points that are ε-far from each other scales doubly exponential in N: DN 1 • Conclusion: all physical states live on a tiny submanifold in Hilbert space; there is no way random states (i.e. following the Haar measure) can be created in nature • What about ground states? Connecting entanglement theory with strongly correlated quantum systems • Strongly correlated quantum systems are at forefront of current experimental research – Cfr. Realization of Mott insulator versus superfluid phase transition in optical lattices (Bloch et al.) – Building of universal quantum simulators using e.g. ion traps – No good theoretical understanding yet: main bottleneck is simulation of quantum Hamiltonians • Quantum spin systems form perfect playground for investigating strongly correlated quantum systems: – Heisenberg model was put forward by Dirac and Heisenberg already in the ’20s as candidate Hamiltonian describing magnetism – Fermi-Hubbard model is believed to be minimal model exhibiting features of high Tc superconductivity (reduces to Heisenberg in some limit) – However, still many open questions! • Quantum spin models arise naturally in the study of quantum error correcting codes – Q.E.C led Kitaev to introducing quantum spin model exhibiting new exotic phases of matter (topological quantum order) – Intriguing connection between ideas in quantum information and condensed matter (e.g. cluster states and valence bond states, …) • What are the questions we would like to see answered? – Ground state properties, energy spectrum, correlation length, criticality, connection between those and entanglement – Are such systems useful, i.e. do they exhibit the right kind of entanglement and allow for the right kind of control, for building e.g. quantum repeaters, quantum memory or quantum computers? – Finite-T: what kind of quantum properties survive at finite T? – Connection between amount of entanglement present in system and simulatability on a classical computer? – Computational complexity of finding ground states? – Dynamics: how much entanglement can be created by local Hamiltonian evolution? • We already have partial answers to those questions: – connection between spectrum and correlation length – criticality in 1-D is accompanied by diverging block entropy. Not such a signature in 2-D (PEPS) – Entanglement length in spin systems versus quantum repeaters – Cluster state quantum computation of Raussendorf and Briegel (cfr. PEPS) – Kitaev: using Toric Code states as fault-tolerant quantum memory in 4-D – Finite T: strict area law for mutual information – MPS/PEPS parameterize manifold of ground states of local Hamiltonians – Kitaev: finding ground states of disordered local Hamiltonians is QMA-complete (also: famous N-representability problem) – Dynamics: Lieb-Robinson bounds Quantum spin systems • Provide perfect playground for investigating nature of entanglement in strongly correlated quantum systems – Most pronounced quantum effects arise at low temperature as large quantum fluctuations exist (ground states) – We assume some geometry and local interactions (cfr. Causality) such as Heisenberg model • Ground states of local spin Hamiltonians are very special: H S x S xj S y S yj S zi S zj i i i, j – Translational invariance implies that energy is completely determined by n.n. reduced density operator ρ of 2 spins: E N . Tr H ij – Finding ground state energy is equivalent to maximizing E over all possible ρ arising from states with the right symmetry – The extreme points of the convex set {ρ} therefore correspond to ground states: ground states are completely determined by their reduced density operators! e.g.: H S x S xj S y S yj S zi S zj i i i, j The Hamiltonian defines hyperplanes in this convex set; convex set is parameterized as 2 x z E () 0 In infinite dimensions: only unentangled states are compatible with symmetry, hence mean-field theory becomes exact (Based on De-Finetti theorem: R. Werner ’89) singlet • Difficulty in characterizing this convex set is due to monogamy / frustration properties of entanglement: a singlet cannot be shared – Entanglement theory allows to make quantitative statements • If local properties of a ground state of a system with N spins and a gap Δ are well approximated, then also the global ones: GS 2 N if i, j : O GS O GS O nn ij ij fermionic systems vs. spin systems • Fundamental question: are fermions fundamentally different from bosons/spins or can local fermionic Hamiltonians be understood as effective Hamiltonians describing low energy sector of specific local spin systems? • Hilbert space associated to fermions is Fock space, which is obtained via second quantization: a ... ci1i2 ... a1 i1 i2 2 i1i2 ... • What we want to approximate is ci1i2 ... • Effective Hamiltonian for this tensor is obtained by doing the Jordan-Wigner transformation on the original one (note the ordering of the fermions in second quantization) • Consider hopping terms in 2-D: J-W induces long-range correlations • Solution: use auxiliary Majorana fermions to turn this Hamiltonian into a local Hamiltonian of spins (cfr. Kitaev) • Similar but different trick applies to any geometry/dimension and multi- channel impurity problems 1 2 3 4 5 10 9 8 7 6 11 12 13 14 15 20 19 18 17 16 Vertical hopping terms become nonlocal by JW-transformation: 9 z x y 9 y a a a a k 10 1 kz 10 † 1 10 † 10 1 JW 1 x k 2 k 2 Solution: add ancillary chains of free fermions bi constructed as follows: define Majorana fermions ci = bi + bi† , di = i(bi - bi†) and free Hamiltonian H anc ick dl k ,l As all terms ic k d l are constants of motion (i.e. +1) and commute with each other, we can change the original vertical hopping terms a1 a10 a10a1 † † a1† a10 a10a1 ic1d10 without changing † the physics of the Hamiltonian. Renumbering everything makes everything local after the JW Entanglement, correlations, area laws Quantifying the amount of correlations between A and B: mutual information B I AB S A S B S AB All thermal states exhibit an exact area law (as contrasted to volume law) A AB exp H F A B Tr H A B S A B Tr H AB S AB I AB 1 Tr H A B AB 1 Tr H AB A B AB Cirac, Hastings, FV, Wolf Similar results for ground states (critical systems might get logarithmic corrections) This is very ungeneric: entanglement is localized around the boundary This knowledge is being exploited to come up with variational classes of states and associated simulation methods that capture the physics needed for describing such systems: * Matrix Product States, Projected Entangled Pair States, MERA Area laws • Main picture: in case of ground states, entanglement is concentrated around the boundary cc Gapped : S 1, 2,, L ln ... 6 cc 1 Critical : S 1, 2,, L 1 ln L ... 12 Kitaev, Vidal, Cardy, Korepin, … Gapped : S 1, 2,, L2 a. L Critical : Free fermions : S 1, 2,, L2 a.L ln L ... Wolf, Klich Critical spin : S 1, 2,, L2 a.L ... quant-ph/0601075 Topological entropy: detects topological quantum order locally! S ABC S AB S AC S BC S A S B S C Kitaev, Preskill, Levin, Wen Ground states of spin Systems • Ground states of gapped local Hamiltonians have a finite correlation length: l OAOB OA OB exp AB C A B C l AB C • Let’s analyze this statement from the point of view of quantum information theory, assuming that AB A B – There is a separable purification of ρAB , so there exists a unitary in region C that disentangles the two parts – Blocking the spins in blocks of log(ξC) spins, then we can write the state as: ABC U , ,il ,ir il ir l il ir r – Doing this recursively yields a matrix product state: i i ... i 12 N A i1 Ai2 ... AiN i1 i2 ... iN Matrix Product States (MPS) D I i i i 1 Map P : H D H D H d State is defined on a d N dimensiona l Hilbert space • Generalizations of AKLT-states (Finitely correlated states, Fannes, Nachtergaele, Werner ‘92) • Gives a LOCAL description of a multipartite state • Translational invariant by construction • Guaranteed to be ground states of gapped local quantum Hamiltonians • The number of parameters scales linearly in N (# spins) • The set of all MPS is complete: Every state can be represented as a MPS as long as D is taken large enough • The point is: if we consider the set of MPS with fixed D, their reduced density operators already approximate the ones obtained by all translational invariant ones very well (and hence also all possible ground states) • MPS have bounded Schmidt rank D • Correlations can be calculated efficiently: contraction of D2x D2 matrices • Numerical renormalization group method of Wilson and Density matrix renormalization group method of S. White can be reformulated and improved upon as variational methods within class of MPS Convex set of reduced density operators of ground states of XXZ-chains approximated with MPS of D=1,2 • So how good will MPS approximate ground states? We want find a bound on the scaling of D as a function of the precision desired and the number of spins N – We impose ex D N N with ε independent of N,D – Because the scaling of the α-entropy of blocks of L spins in spin chains is bounded by cc 1 S 1 1 ln Tr 1 ln L 12 cst it follows that it is enough to choose DN N f (c ) FV, Cirac – It shows D only has to grow as a polynomial in the number of particles to obtain a given precision, even in the critical case! • M. Hastings (2007): – All ground states of gapped Hamiltonians are well represented by MPS because they obey an area law – Same proof in principle applies to the higher dimensional generalizations of MPS: PEPS • MPS / PEPS are hence the ideal variational class of wavefunctions for simulating strongly correlated quantum spin systems; in other words: we have identified the right submanifold! • What about the complexity of finding this optimal MPS in the worse case? – Finding ground state of a local 1-D quantum spin chain with a gap that is bounded below by c||H||/poly(N) is NP-hard – Proof goes via identifying a family of such Hamiltonians that is NP- complete and have ground states that are exactly matrix product states • Sketch of proof: cfr. Aharonov, Gottesman, Kempe proof of QMA-hardness of finding GS of 1-D quantum spin systems, but use classical circuit instead Ground state of corresponding Hamiltonian is of the form t t t t ooo... o o o o... t Nt N (T t 1) – As all t U tU t 1...U1 0 are classical, this ground state has very few entanglement; in fact it is a MPS with dimension poly(N) and hence checking energy is in NP; finding MPS is hence NP-complete – Gap of Hamiltonian comes from random walk – We can also construct alternative Hamiltonians starting from classical / quantum reversible cellular automata Schuch, Cirac, FV Wilson’s numerical renormalization group • Consider Kondo-impurity-like problem with Hamiltonian N k x x 1 y y 1 k k k k k 0 A i1 i2 2 i2 i1 i1 ,i2 3 A 2 i3 i 3 ,i3 1 2 3 4 5 6 N A A A ...A i1 ,i2 ,... i2 i1 i3 i4 iN i1 i2 ... iN A i1 ,i2 ,... i2 Ai3 Ai4 ... AiN i1 i2 ... iN , ,... • NRG method creates an effective Hamiltonian which is the original Hamiltonian projected in a basis of matrix product states (MPS) • Success of NRG follows from the fact that those MPS parameterize well the low-energy sector of the Hilbert space • Main new ingredient from DMRG: sweep! S. White’s DMRG method • Extending DMRG to periodic boundary conditions: P1 P2 P3 P4 P5 PN -4 DMRG (PBC) 10 DM E 0 /|E0| RG < m i S i+1> / E0 - 1 -6 10 (O BC Ne ) -8 w 10 (P BC S ) mi -10 10 0 20 40 60 Variational dimensional reduction of MPS • Given a D-dimensional D MPS parameterized by the DxD matrices Ai, find D ' parameterized by D’xD’matrices Bi (D’< D) such as to minimize i i Tr B1i B1i B2 B2 BN BN 2 i i i i i i i 2Tr B1i A1i B2 A2 BN AN cst i i i i i – Can be minimized variationally by iteratively solving linear systems of equations • This can be used to describe both real and imaginary time-evolution Generalizations of MPS • PEPS: 2-dimensional generalization of MPS • MPS and weighted graph states (Briegel, Dur, Eisert, Plenio) • MERA: multiscale entanglement renormalization ansatz (Vidal) – Allows to represent critical or scale-invariant wavefunctions – Can be created using a tree-like quantum circuit of unitaries and isometries Generalizations of MPS to higher dimensions • The MPS/AKLT picture can be generalized to any geometry : Projected Entangled Pair States (PEPS) P maps D4 dimensional to d dimenional space • Properties: Area Law automatically fulfilled; local properties can be approximated very well ; guaranteed to be ground states of local Hamiltonians; again, every state can be written as a PEPS • A variational approach based on those states provides a solution to the problem of the numerical renormalization group approach where the number of degrees of freedom of a block grows exponentially with the size of the block • How to calculate correlation functions? – Instead of contracting matrices, we have to contract tensors: i0 X aX aX aX a a aX aX aX aX i1 a a a a X a 1 1 2 2 3 3 a a 4 4 5 5 6 6 7 7 8 8 i M M 1 XX X X X X X X X i 2 i2 M XX X X X X X X X i3 i M 3 XX X X X X X X X i4 i M 4 XX X X X X X X X i5 i5 M XX X X X X X X X iM0 V. Murg, FV, I. Cirac Examples of PEPS • Cluster states of Briegel and Raussendorf are PEPS with D=2: allow for universal quantum computation with local measurements only. We can also construct other states that are universal using PEPS • PEPS with topological quantum order: – Toric code states of A. Kitaev (D=2): fault-tolerant quantum memory – Resonating valence bond states (D=3) • PEPS with D=2 can be critical: power law decay of correlations – Many examples can be constructed by considering coherent versions of classical statistical models: exp i j 1 2 N 1 2 N 2 i, j – Resolves open question about scaling of entanglement in critical 2-D quantum spin systems: no logarithmic corrections – PEPS construction shows that for every classical temperature-driven phase transition there exists a quantum spin model in the same dimension exhibiting a zero-T quantum phase transition with same features • PEPS provide perfect playground for considering open questions like existence of deconfined criticality: all PEPS are ground states Conclusion • Formalism of quantum information theory provides unique perspective on strongly correlated quantum systems – MPS/PEPS picture describes low-energy sector of local Hamiltonians, and opens a whole new toolbox of numerical renormalization group methods that allows to go where nobody has gone before • Similar ideas can be used in context of lattice gauge theories, quantum chemistry, … – Frustration and monogamy properties of entanglement (cryptography), quantum error correction, and the complexity of simulating quantum systems are basic notions in the fields of quantum information and statistical physics – Synergy of quantum information and the theory of strongly correlated quantum systems opens up many new themes for both fields and could lead to a much more transparent description of the whole body of quantum physics Work described is mainly from: I. Cirac, J. Garcia-Ripoll, M. Martin-Delgado, V. Murg, B. Paredes, D. Perez-Garcia, M. Popp, D. Porras, C. Shon, E. Solano, M. Wolf (Max Planck Institute for Quantum Optics), J. von Delft, A. Weichselbaum (LMU), U. Schollwock (RWTH), M. Hastings, G. Ortiz (Los Alamos), T. Osborne (London U)