Many-Body physics meets Quantum Information

Many-Body physics meets Quantum Information Rosario Fazio Scuola Normale Superiore, Pisa & NEST, Istituto di Nanoscienze - CNR, Pisa

Quantum Computers Interaction between qubits two-level systems Many-Body Systems Controlled in the... - preparation - evolution - measurement

Links between Quantum Information & Statistical Mechanics - Quantum Information tools in Condensed Matter - New methods to study Many-Body problems - Adiabatic quantum comp. vs Kibble-Zurek mechanism

Links between Quantum Information & Statistical Mechanics - Quantum Information tools in Condensed Matter - New methods to study Many-Body problems - Adiabatic quantum comp. vs Kibble-Zurek mechanism Review: L. Amico, et al Rev.Mod.Phys. 80, 517 (2008)

Entanglement If Ψ ab ψ a χ b then the state is entangled ENTANGLEMENT as a RESOURCE It is believed to be the main ingredient of computational speed-up in quantum information protocols

Entanglement in Condensed Matter Spin Systems Superconductivity Quantum Hall Effect. - Characterization of condensed phases - Collective phenomena in Quantum Information and Quantum Communication

How to measure entanglement - Entanglement between two spins in the network (bipartite) - Multipartite entanglement - Block entropy - Localizable entanglement -...

Bipartite entanglement Correlation ρ(i, j) The state of the two selected spins is mixed

ξ (g c g) ν order parameter (g c g) β g

Ising chain in a transverse field H = J 2 N (1 γ)σi x σi+1 x +(1+γ)σ y N i σy i+1 h i=1 i=1 σ z i σ x 0 1 λ = J 2h Exact solution - free fermions

Next neighbor entanglement critical region

Block entropy Ground State: Ψ GS A B A L sites S(ρ L ) tr(ρ L logρ L ) ρ L = tr N L ( Ψ GS Ψ GS )

Block entropy S(ρ L ) tr(ρ L logρ L ) A B A S L c 6 log 2 L critical chain S L c 6 log 2 ξ non-critical chain Area s law in d dim S B L d 1

Entropy scaling 1.6 1.1 1.08 1.4 c 1.06 1.04 1.02 S l 1.2 1 10 3 10 2 1/N 1.04 1.02 1 c 1 0.98 0.8 0.96 0.5 0 0.5 1 1 10 100 l

Adiabatic quantum computation H f Easy to write E. Fahri, J. Goldstone and S. Gutmann 00 The ground state may still be very difficult to find σ 1 = σ 2 = σ 3 σ 25 = σ 32 = σ 12 σ 16 = σ 44 = σ 1 h ijk = σ z i σ z j σ z k +

Adiabatic quantum computation H i E. Fahri, J. Goldstone and S. Gutmann 00 The ground state is known H f The ground state is the solution to our problem Adiabatic evolution H(t) = T t T H i + t T H f

Adiabatic quantum computation E 1 T 1 2 m m 1 N η Easy problem m m e N η Difficult problem E G Quantum speed up Quantum critical point

Topological Defect Formation Signatures of phase transitions which have occurred in the early universe by determining the density of defects left in the broken symmetry phase as a function of the rate of quench. Kibble 76, Zurek 85

Topological Defect Formation Simulation of phase transitions in the early universe in condensed matter systems (superfluids and Josephson junctions) Th: Zurek 85-88 Exps:Bauerle et al 96,Ruutu et al 96} Extension to quantum phase transitions Zurek, Dorner, Zoller 05 Polkovnikov 05...

Adiabatic Dynamics Close to a Critical Point τ! (b)!t^ λ c ^ t λ(t) t How effective is it to execute a given computational task by slowly varying in time the Hamiltonian of a quantum system? Is it possible to find the ground state of a classical system by slowly annealing away its quantum fluctuations? What is the density of defects left over after a passage through a continuous (quantum) phase transition?

Defect density W. Zurek 85 W. Zurek, U. Dorner and P. Zoller 05 A. Polkovnikov 05 " H = H 0 + (a) λh 1 λ λ c = vt!t^ ^ t t The adiabatic approximation breaks down when λ λ τ! (b) ρ def ˆξ d v dν zν+1!t^ ^ t t E res Jρ def

1D Ising Model N { σ x j σj+1+h(t)σ x j z } H = J 2 X j v 0 h c =1 E res v

Quantum Critical Region Incoherent production

Incoherent Defects Density of defects E E KZ + E inc The bath does not influence the system for T Relaxation in the critical region τ 1 r αt θ E = d d k (2π) d P k t QC =2T 1/νz v 1 d dt P k = 1 τ Pk P th k (h c )

Incoherent Defects E inc αv 1 T θ+ dν+1 νz v cross α ( νz+1 1+ (θ 1)νz ν(z+d)+1 T ν(z+d)+1 ) ( 1+ 1 νz)

Ψ >= {i α } C(i 1,i 2,...,i N ) i 1,i 2,.i N > i α =1,...,d......... i 1 i 2 i N Exponential Number of Parameters!

Variational Approach Educated guess of the ground state wavefunction EXAMPLE: Gutzwiller approximation of the Bose-Hubbard Model H = U i n i (n i 1) µ i n i t <ij> a i a j κ Ψ >= i n i e κ(ni n) n i >

Variational ansatzs & Quantum Information Briegel, Cirac, Eisert, Hastings, Latorre, Plenio, Verstraete,Vidal,... New insight on variational wave-functions from Quantum Information - General schemes for efficient computation of a variational functions (MPS, PEPS, MERA,...) - Account for entanglement properties (crucial for critical systems) - Extensions to time-dependent situations, finite temperatures,...

Multiscale Entanglement Renormalization Ansatz Ψ >= {i α } C(i 1,i 2,...,i N ) i 1,i 2,.i N > G. Vidal (2005-2009)...... i 1 i 2 i N

MERA: Definitions u1 l1 u2 l2 G. Vidal (2005-2009) u1 l1 l2 l1 l2 l3 l4 = =

Real Space Renormalization N=4 N=8 N=16 Ψ im (M sites) Ψ im +1(M/2 sites)

Causal Cone

Critical exponents Ising Model H = j Jσ x j σ x j+1 + j B σ z j < σ α σ α > < σ α >< σ α > α ν exact α ν num α x 0.25 0.2509 0.36 % y 2.25 2.2544 0.19 % z 2 2.0939 4.48 % ε

Links between Quantum Information & Statistical Mechanics - Quantum Information tools in Condensed Matter - New methods to study Many-Body problems - Adiabatic quantum comp. vs Kibble-Zurek mechanism