Area law and Quantum Information José Ignacio Latorre Universitat de Barcelona Cosmocaixa, July 2006

Bekenstein-Hawking black hole entropy Entanglement entropy A B

Entropy sets the limit for the simulation of QM Goal of the talk Area law in QFT PEPS in QI

Schmidt decomposition A B  =min(dim H A, dim H B ) is the Schmidt number Some basics

The Schmidt number measures entanglement Let’s compute the von Neumann entropy of the reduced density matrix  =1 corresponds to a product state Large  implies large number of superposed states A B Srednicki ’93:

Maximally entangled states (EPR states) Each party is maximally surprised when ignoring the other one 1 ebit Ebits are needed for e.g. teleportation (Hence, proliferation of protocoles of distillation)

Maximum Entropy for N-qubits Strong subadditivity implies concavity

U entanglement preparation evolution measurement quantum computer simulation Quantum computation How accurately can we simulate entanglement ?

Exponential growth of Hilbert space Classical representation requires d n complex coefficients n A random state carries maximum entropy

Efficient description for slightly entangled states A B  = min(dim H A, dim H B ) Schmidt number Back to Schmidt decomposition A product state corresponds to

Slight entanglement iff  poly(n)<< d n Representation is efficient Single qubit gates involve only local update Two-qubit gates reduces to local updating Vidal: Iterate this process A product state iff efficient simulation

Small entanglement can be simulated efficiently quantum computer more efficient than classical computer if large entanglement

Matrix Product States i α Approximate physical states with a finite  MPS canonical form

Graphic representation of a MPS Efficient computation of scalar products operations

Intelligent way to represent entanglement!! Ex: retain 2,3,7,8 instead of 6,14,16,21,24,56 Efficient representation Efficient preparation Efficient processing Efficient readout

Matrix Product States for continuous variables Harmonic chains MPS handles entanglementProduct basis Truncate  tr d tr Iblisdir, Orús, JIL

Nearest neighbour interaction Minimize by sweeps (periodic DMRG, Cirac-Verstraete) Choose Hermite polynomials for local basis optimize over a

Results for n=100 harmonic coupled oscillators (lattice regularization of a quantum field theory) d tr =3  tr =3 d tr =4  tr =4 d tr =5  tr =5 d tr =6  tr =6 Newton-raphson on a

Quantum rotor (limit Bose-Hubbard) Eigenvalue distribution for half of the infinite system

Simulation of Laughlin wave function Local basis: a=0,..,n-1 Analytic expression for the reduced entropy Dimension of the Hilbert space

Exact MPS representation of Laughlin wave function Clifford algebra Optimal solution! (all matrices equal but the last)

m=2

Spin-off? Problem: exponential growth of a direct product Hilbert space Computational basis MPS Neural network i1i1 i2i2 inin

MPS Product states H NN Non-critical 1D systems ?

i 1 =1 i 1 =2 i 1 =3 i 1 =4 | i 1  i 2 =1 i 2 =2 i 2 =3 i 2 =4 | i 2 i 1  105| 2,1  Spin-off 1: Image compression pixel addresslevel of grey RG addressing

QPEG Read image by blocks Fourier transform RG address and fill Set compression level:  Find optimal gzip (lossless, entropic compression) of (define discretize Γ’s to improve gzip) diagonal organize the frequencies and use 1d RG work with diferences to a prefixed table Low frequencies high frequencies

 = 1 PSNR=17  = 4 PSNR=25  = 8 PSNR=31 Max  = 81

Spin-off 2: Differential equations Good if slight correlations between variables

Limit of MPS 1D chains, at the quantum phase transition point : scaling Quantum Ising, XY c=1/2 XX, Heisenberg c=1 Universality Away from criticality: saturation MPS are a faithful representation for non-critical 1D systems but deteriorate at quantum phase transitions Vidal, Rico, Kitaev, JIL Callan, Wilczeck

Exact coarse graining of MPS Optimal choice! VCLRW  remains the same and locks the physical index! After L spins are sequentially blocked Entropy is bounded Exact description of non-critical systems Local basis

Area law for bosonic field theory Geometric entropy Fine grained entropy Entanglement entropy QFT geometry

Radial discretization Srednicki ‘93

+ lots of algebra

Area Law for arbitrary dimensional bosonic theory Riera, JIL Vacuum order: majorization of renduced density matrix Eigenvalues of  Majorization in L: area law Majorization along RG flows

Majorization theory Entropy provides a modest sense of ordering among probability distributions Muirhead (1903), Hardy, Littlewood, Pólya,…, Dalton Consider such that p are probabilities, P permutations d cumulants are ordered D is a doubly stochastic matrix LL  L t  L t’ t t’ RG Vacuum reordering

Area law and gravitational anomalies c 1 is an anomaly!!!! Von Neumann entropy captures a most elementary counting of degrees of freedom Trace anomalies Kabat – Strassler

Is entropy coefficient scheme dependent is d>1+1? Yes No c 1 =1/6 bosons c 1 =1/12 fermionic component

A B S A = S B → Area Law Entanglement bonds Contour (Area) law S ~ n (d-1)/d Can we represent an Area law? Locality symmetry

Efficient singular value decomposition BUT ever growing Area Law and RG of PEPS P rojected E ntangled P air PEPS can support area law!!

Can we handle quantum algorithms?

Adiabatic quantum evolution Farhi-Goldstone-Gutmann H(s(t)) = (1-s(t)) H 0 + s(t) H p Inicial hamiltonianProblem hamiltonian s(0)=0 s(T)=1 t Adiabatic theorem: if E1E1 E0E0 E t g min

3-SAT –3-SAT 3-SAT is NP-complete K-SAT is hard for k > SAT with m clauses: easy-hard-easy around m=4.2 –Exact Cover A clause is accepted if 001 or 010 or 100 Exact Cover is NP-complete For every clause, one out of eight options is rejected instance

Beyond area law scaling! n=6-20 qubits 300 instances n/2 partition S ~.1 n Orús-JIL entropy s

n=80 m=68  =10 T=600 Max solved n=100 chi=16 T=5000

New class of classical algorithms: Simulate quantum algorithms with MPS Shor’s uses maximum entropy with equidistribution of eigenvalues Adiabatic evolution solved a n=100 Exact Cover! 1 solution among 10 30

Non-critical spin chainsS ~ ct Critical spin chainsS ~ log 2 n Spin chains in d-dimensions (QFT) S ~ n (d-1)/d Violation of area law!! (some 2D fermionic models) S ~ n 1/2 log 2 n NP-complete problemsS ~.1 n Shor FactorizationS ~ r ~ n Summary

Beyond area law? VIDAL: Entanglement RG Multiscale Entanglement Renormalization group Ansatz

Simulability of quantum systems QPT MERA? PEPS finite  Physics ? QMA ? Area law MPS

Quantum Mechanics Classical Physics + classification of QMA problems!!!