Multipartite Entanglement Measures from Matrix and Tensor Product States

1. Multipartite Entanglement Measures from Matrix and Tensor Product States Ching-Yu Huang Feng-Li Lin Department of Physics, National Taiwan Normal University arXiv:0911.4670

2. Outline • Quantum Phase Transition • Entanglement measure • Matrix/Tensor Product States • Entanglement scaling • Quantum state RG transformation

3. Quantum Phase transition • Phase transition due to tuning of the coupling constant. • How to characterize quantum phase transition? a. Order parameters

4. Frustrated systems • These systems usually have highly degenerate ground states. • This implies that the entropy measure can be used to characterize the quantum phase transition. • Especially, there are topological phases for some frustrated systems (no order parameters) ?

5. Entanglement measure • The degeneracy of quantum state manifests as quantum entanglement. • The entanglement measure can be used to characterize the topological phases. • However, there is no universal entanglement measure of many body systems. • Instead, the entanglement entropy like quantities are used.

6. Global entanglement • Consider pure states of N particles . The global multipartite entanglement can be quantified by considering the maximum fidelity. • The larger indicates that the less entangled. • A well-defined global measure of entanglement is 【Wei et al. PRA 68, 042307(2003)】

7. Global entanglement density and its h derivative for the ground state of three systems at N. Ising ; anisotropic XY model; XX model.

8. Our aim • We are trying to check if the entanglement measure can characterize the quantum phase transition in 2D spin systems. • Should rely on the numerical calculation • Global entanglement can pass the test but is complicated for numerical implementation. • One should look for more easier one.

9. Matrix/Tensor Product States • The numerical implementation for finding the ground states of 2D spin systems are based on the matrix/tensor product states. • These states can be understood from a series of Schmidt (bi-partite) decomposition. It is QIS inspired. • The ground state is approximated by the relevance of entanglement.

10. » 2 N # parameters NdD d << 1D quantum state representation Virtual dimension = D 【Vidal et al. (2003)】 Physical dimension = d λN-1 λ2 λ1 Γ1 Γ 2 Γ N Γ3 • Representation is efficient • Single qubit gates involve only local update • Two-qubit gates reduces to local updating

11. ``Solve” for MPS/TPS • In old days, the 1D spin system is numerically solved by the DMRG(density matrix RG). • For translationally invariant state, one can solve the MPS by variational methods. • More efficiently by infinite time - evolving block decimation (iTEBD) method.

12. Time evolution • Real time • imaginary time • Trotter expansion G F F G λ1 λ2 λ3 λ2 λ4 λ3 λ4 λ1 Γ1 Γ3 Γ5 Γ3 Γ1 Γ 2 Γ 2 Γ4 Γ4 Γ5 【Vidal et al. (2004)】

13. Evolution quantum state Θ U λi λ’k λi λk+1 λk+1 λ'k λk λk+1 Γ(sj) λk+1 X(si) Γ’(si) Γ (si) Γ’(sj) Y (sj) λi λ-1i SVD

14. Global entanglement from matrix product state for 1D system • Matrix product state (MPS) form • The fidelity can be written by transfer matrix where B1 A1 A3 B3 A2 B2 AN BN 【Q.-Q. Shi, R. Orus, J.-O. Fjaerestad, H.-Q. Zhou, arXiv:0901.2863 (2009)】

15. Quantum spin chain and entanglement • XY spin chain • Phase diagram: oscillatory (O) ferromagnetic (F) paramagnetic (P) 【Wei et al. PRA 71, 060305(2003)】

16. A second-order quantum phase transition as h is tuned across a critical value h = 1. • Magnetization along the x direction . (D= 16)

17. For a system of N spin 1/2.the separable statesrepresented by a matrix • The global entanglement has a nonsingular maximum at h=1.1. • The von Neuman entanglement entropy also has similar feature.

18. Generalization to 2D problem • The easy part promote 1D MPS to 2D Tensor product state (TPS) • The different partHow to perform imaginary time evolutiona. iTEBDHow to efficiently calculate expectation valuea. TRG 【Levin et al. (2008), Xiang et al. (2008) , Wen et al. (2008) 】

19. 2D Tensor product state (TPS) • Represent wave-function by the tensor network of T tensors u Virtual dimension = D l r p d Physical dimension = d

20. Global entanglement from matrix product state for 2D system Double Tensor • The fidelity can be written by transfer matrix where • It is difficult to calculate tensor trace (tTr), so we using the “TRG” method to reduce the exponentially calculation to a polynomial calculation. B1 B2 Ť1 A2 A1 Ť2 B4 B3 A4 A3 Ť4 Ť3

21. S3 u l γ r T T S1 d u S4 T T r γ l S2 d The TRG methodⅠ • First, decomposing the rank-four tensor into two rank-three tensors.

22. γ2 γ1 γ1 γ2 a1 T’ S1 S2 a2 a4 S4 S3 a3 γ3 γ3 γ4 γ4 The TRG methodⅡ • The second step is to build a new rank-four tensor. • This introduces a coarse-grained square lattice.

23. The TRG methodⅢ • Repeat the above two steps, until there are only four sites left. One can trace all bond indices to find the fidelity of the wave function. TA TB TC TD

24. Global entanglement for 2D transverse Ising • Ising model in a transverse magnetic field • The global entanglement density v.s. h, the entanglement has a nonsingular maximum at h=3.25. • The entanglement entropy

25. Global entanglement for 2D XXZ model • XXZ model in a uniform z-axis magnetic field • A first-order spin-flop quantum phase transition from Neel to spin-flipping phase occurs at some critical field hc. • Another critical value at hs = 2(1 + △), the fully polarized state is reached.

26. L spins ρL SL Scaling of entanglement Entanglement entropy for a reduced block in spin chains 【 Vidal et al. PRL,90,227902 (2003)】 At Quantum Phase Transition Away from Quantum Phase Transition

27. Block entanglement v.s. Quantum state RG • Numerically, it is involved to calculate the block entanglement because the block trial state is complicated. • Entanglement per block of size 2L= entanglement per site of the L-th time quantum state RG(merging of sites). • We can the use quantum state RG to check the scaling law of entanglement.

28. RG γ α γ β α RG p q l 1D Quantum state transformation Quantum state RG 【Verstraete et al. PRL 94, 140601 (2005)】 • Map two neighboring spins to one new block spin • Identify states which are equivalent under local unitary operations. By SVD

29. For MPS representation for a fixed value of D, this entropy saturates at a distance L～Dκ(kind of correlation length) • Could we attain the entropy scaling at critical point from MPS? At quantum point Near quantum point 【 Latorre et al. PRB,78,024410 (2008)】

30. MPS support of entropy obeys scaling law!!

31. 2D Scaling of entanglement • In 2D system, we want to calculate the block entropy and block global entanglement. Could we find the scaling behavior like 1D system? Area law

32. s1 s2 RG s s4 s3 RG 2D Quantum state transformation • Map one block of four neighboring spins to one new block spin. • By TRG and SVD method

33. The failure • We find that the resulting block entanglement decreases as the block size L increases. • This failure may mean that we truncate the bond dimension too much in order to keep D when performing quantum state RG transformation, so that the entanglement is lost even when we increase the block size.

34. Summary • Using the iTEBD and TRG algorithms, we find the global entanglement measure.- 1D XY model- 2D Ising model- 2D XXZ model • The scaling behaviors of entanglement measures near the quantum critical point.

35. Thank You For Your Attention!