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

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

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

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) ?

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.

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, (2003) 】

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

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.

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.

1D quantum state representation Γ N Γ1Γ1 Γ 2 Γ3Γ3 λ1λ1 λ2λ2 λ N-1 N dNdD parameters <<  2 # Representation is efficient Single qubit gates involve only local update Two-qubit gates reduces to local updating Virtual dimension = D Physical dimension = d 【 Vidal et al. (2003) 】

``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.

Time evolution Real time imaginary time Trotter expansion Γ1Γ1 Γ 2 Γ3Γ3 λ1λ1 λ2λ2 Γ4Γ4 Γ5Γ5 λ3λ3 λ4λ4 F F Γ1Γ1 Γ3Γ3 λ1λ1 λ2λ2 Γ4Γ4 Γ5Γ5 λ3λ3 λ4λ4 G G 【 Vidal et al. (2004) 】

Γ (si) Γ (sj) λiλi λkλk λ k+1 U Θ SVD Γ’ (si) Γ’ (sj) λiλi λ’kλ’k λ k+1 X (si) Y (sj) λ'kλ'k λ -1 i λiλi λ k+1 Evolution quantum state

Global entanglement from matrix product state for 1D system Matrix product state (MPS) form The fidelity can be written by transfer matrix where 【 Q.-Q. Shi, R. Orus, J.-O. Fjaerestad, H.-Q. Zhou, arXiv: (2009) 】 ANAN A1A1 A2A2 A3A3 BNBN B1B1 B2B2 B3B3

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

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

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

Generalization to 2D problem The easy part promote 1D MPS to 2D Tensor product state (TPS) The different part How to perform imaginary time evolution a. iTEBD How to efficiently calculate expectation value a. TRG 【 Levin et al. (2008), Xiang et al. (2008), Wen et al. (2008) 】

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

Global entanglement from matrix product state for 2D system 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. A4A4 A3A3 A1A1 A2A2 B1B1 B4B4 B2B2 B3B3 Ť4Ť4 Ť2Ť2 Ť3Ť3 Ť1Ť1 Double Tensor

The TRG method Ⅰ First, decomposing the rank-four tensor into two rank-three tensors. u S2S2 d S4S4 S3S3 l u S1S1 T T T T r d l γ γ r

The TRG method Ⅱ The second step is to build a new rank-four tensor. This introduces a coarse-grained square lattice. γ1γ1 γ2γ2 γ3γ3 T’T’ γ4γ4 γ1γ1 S1S1 S4S4 S3S3 S2S2 a1a1 a2a2 a3a3 a4a4 γ2γ2 γ3γ3 γ4γ4

The TRG method Ⅲ 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. TATA TCTC TBTB TDTD

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

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.

Scaling of entanglement L spins ρLρL SLSL 【 Vidal et al. PRL,90, (2003) 】 Entanglement entropy for a reduced block in spin chains At Quantum Phase TransitionAway from Quantum Phase Transition

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 2 L = 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.

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

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

MPS support of entropy obeys scaling law!!

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

2D Quantum state transformation Map one block of four neighboring spins to one new block spin. By TRG and SVD method RG s1s1 s2s2 s4s4 s3s3 s

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.

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.

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