Entanglement, quantum critical phenomena and efficient simulation of quantum dynamics

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1 Entanglement, quantum critical phenomena and efficient simulation of quantum dynamics Simons Conference on Reversible and Quantum Computation Stony Brook, May Guifre Vidal Institute for Quantum Information, Caltech

2 outline Quantum Information ENTANGEMENT Entanglement in quantum phase transitions Quantum Many-Body Physics Efficient classical simulation of quantum dynamics Critical and non-critical spin chains. Non-critical spin lattices in 2D, 3D. Scaling of entanglement in critical and non-critical spin chains. Emergence of universality at a quantum critical point. Connection to conformal field theory, irreversibility of RG flow. Entanglement in spin lattices. Failure of the DMRG method.

3 Entanglement in quantum phase transitions Vidal, atorre, Rico, Kitaev, quant-ph/ (to appear in Phys. Rev. ett.) atorre, Rico, Vidal, quant-ph/ Measures of entanglement in a quantum spin chain T=0, ground state concurrence (entanglement between two spins) Osterloh, Amico, Falci and Fazio, Nature 2002 Osborne and Nielsen, Phys. Rev. A 2002 Our approach: spins entropy S of a block of spins (Entanglement between block of spins and rest of the chain)

4 Entanglement in quantum phase transitions XY model with magnetic field [including XX model and Ising model] H XY = N 1 l= 0 (1 + γ ) σ 2 x l σ (1 γ ) 2 x y y z l+ 1 + σ l σ l+ 1 + λσ l Ground state: Ψ g XY model: ieb, Schultz and Mattis, Ann. Phys. (1961) XY model with magnetic field: Barouch and McCoy, Phys. Rev. A (1971) gaussian in fermionic modes (efficient description) ρ S = tr( ρ ) 1 ρ 2 S2 ρ 3 S3 1 1 logρ1 ρ S

5 Entanglement in quantum phase transitions Scaling of entanglement in critical and non-critical spin chains Ising model for different values of the magnetic field λ λ = λ c critical chain 1 S log 6 non-critical S ( ) chains S * λ λ = 0

6 Entanglement in quantum phase transitions Emergence of Universality 1 S log 3 Critical XX ASYMPTOTICS Jin and Korepin, quant-ph Critical XXZ 1 S log 6 Critical Ising Critical XY

7 Entanglement in quantum phase transitions Connection to conformal field theory central charge geometric c + c entropy S log 6 C-theorem Entanglement decreases under RG flow Spin lattices in D>1 dimensions Area law Extra bonus! S D 1 Holzhey, arsen, Wilczek, Nucl. Phys. B (1994) Fiola, Preskill, Strominger, Trivedi, Phys Rev D (1994) Zamolodchikov, JETP ett (1986) Capelli, Friedan, atorre, Nucl. Phys. B (1991) Forte, atorre, Nucl. Phys. B (1998) Srednicki, Phys. Rev. ett. (1993) Srednicki, PR 71 (1993) Failure of White s DMRG numerical method in 2D,3D # of eigenvectors of ρ m 2 S 1D 2D, 3D noncritical critical

8 Efficient classical simulation of quantum dynamics Vidal, quant-ph/ Vidal, in preparation Measure of multipartite entanglement A B Ψ Schmidt decomposition χ A = λ α Φα Φα α = 1 [ A] [ B] χ max χ A A Schmidt rank χ A Only vanishes for product (i.e. unentangled) states E log χ 2 χ Additive under tensor product Non-increasing under OCC (even under SO)

9 Efficient classical simulation of quantum dynamics Decomposition of N-qubitN states standard decomposition Ψ c i 1 i 2 Λ i l Λ i l N N 2 coefficients N N qubits new decomposition Γ [ 1] [ 2 ] [ l ] [ N ] Γ Λ Γ Λ Γ l =1, Λ, i = 0,1 N [l]i Γ αβ α = 1, Λ, χ N exp E χ coefficients β = 1, Λ, χ

10 Efficient classical simulation of quantum dynamics Non-critical spin chain χ χ max χ χ 0 (correlation length) (size of the block) saturation of χ O(N) parameters to describe N spins

11 Efficient classical simulation of quantum dynamics N spins Non-critical 1D system Critical 1D system cost of simulation O(N) q O( N ) q>1 Non-critical 2D system Critical 2D system O( N exp N ) alternative method O(N) Non-critical 3D system Critical 3D system O( N exp N 2/3 ) for non-critical systems

12 summary Entanglement in Quantum Many-Body Physics descriptive Classical simulation of quantum dynamics Critical and non-critical spin chains. Non-critical spin lattices in 2D, 3D. constructive Entanglement in quantum phase transitions Scaling of entanglement in critical and non-critical spin chains. Emergence of universality at a quantum critical point. Conformal field theory Monotonicity under RG flow. 2D,3D systems. Failure DMRG method.