Entanglement, quantum critical phenomena and efficient simulation of quantum dynamics Simons Conference on Reversible and Quantum Computation Stony Brook, May 28-31 2003 Guifre Vidal Institute for Quantum Information, Caltech
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.
Entanglement in quantum phase transitions Vidal, atorre, Rico, Kitaev, quant-ph/0211074 (to appear in Phys. Rev. ett.) atorre, Rico, Vidal, quant-ph/0304098 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)
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
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
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
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
Efficient classical simulation of quantum dynamics Vidal, quant-ph/0211074 Vidal, in preparation Measure of multipartite entanglement A B 1 2 3 4 5 6 7 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)
Efficient classical simulation of quantum dynamics Decomposition of N-qubitN states standard decomposition Ψ c i 1 i 2 Λ i l Λ i 1 2 3 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
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
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
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.