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Area law and Quantum Information

Area law and Quantum Information. José Ignacio Latorre Universitat de Barcelona Cosmocaixa, July 2006. Bekenstein-Hawking black hole entropy. Entanglement entropy. A. B. Goal of the talk. Entropy sets the limit for the simulation of QM. Area law in QFT PEPS in QI.

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Area law and Quantum Information

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

  2. Bekenstein-Hawking black hole entropy Entanglement entropy A B

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

  4. Some basics Schmidt decomposition A B =min(dim HA, dim HB) is the Schmidt number

  5. 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 Srednicki ’93: B

  6. 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)

  7. Maximum Entropy for N-qubits Strong subadditivity implies concavity

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

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

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

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

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

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

  14. Graphic representation of a MPS Efficient computation of scalar products operations

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

  16. Matrix Product States for continuous variables Iblisdir, Orús, JIL Harmonic chains MPS handles entanglement Product basis Truncate tr dtr

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

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

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

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

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

  22. m=2

  23. Spin-off? Problem: exponential growth of a direct product Hilbert space in i2 i1 Computational basis Neural network MPS

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

  25. i2=1 i2=2 i2=3 i2=4 Spin-off 1: Image compression | i2 i1 | i1 105| 2,1  i1=1 i1=2 i1=3 i1=4 RG addressing level of grey pixel address

  26. Low frequencies • 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 high frequencies

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

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

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

  30. Exact coarse graining of MPS Local basis 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

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

  32. Srednicki ‘93 Radial discretization

  33. + lots of algebra

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

  35. Majorization theory Entropy provides a modest sense of ordering among probability distributions Muirhead (1903), Hardy, Littlewood, Pólya,…, Dalton Consider such that pare probabilities,Ppermutations dcumulants are ordered Dis a doubly stochastic matrix Vacuum reordering RG t t’ Lt Lt’ L

  36. Area law and gravitational anomalies c1 is an anomaly!!!! Von Neumann entropy captures a most elementary counting of degrees of freedom Trace anomalies Kabat – Strassler

  37. Is entropy coefficient scheme dependent is d>1+1? Yes No c1=1/6 bosons c1=1/12 fermionic component

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

  39. Efficient singular value decomposition BUT ever growing Area Law and RG of PEPS Projected Entangled Pair PEPS can support area law!!

  40. Can we handle quantum algorithms?

  41. t s(T)=1 H(s(t)) = (1-s(t)) H0 + s(t) Hp s(0)=0 Adiabatic quantum evolutionFarhi-Goldstone-Gutmann Inicial hamiltonian Problem hamiltonian Adiabatic theorem: if E E1 gmin E0 t

  42. 3-SAT • 3-SAT • 3-SAT is NP-complete • K-SAT is hard for k > 2.41 • 3-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 0 1 1 0 0 1 1 0 instance For every clause, one out of eight options is rejected

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

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

  45. Adiabatic evolution solved a n=100 Exact Cover! 1 solution among 1030 New class of classical algorithms: Simulate quantum algorithms with MPS Shor’s uses maximum entropy with equidistribution of eigenvalues

  46. Summary

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

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

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

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