Universal Quantum Simulator, Local Convertibility and Edge States in Many-Body Systems Fabio Franchini

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1 New Frontiers in Theoretical Physics XXXIV Convegno Nazionale di Fisica Teorica - Cortona Universal Quantum Simulator, Local Convertibility and Edge States in Many-Body Systems Fabio Franchini Collaborators: J. Cui, L. Amico, H. Fan, M. Gu, V. E. Korepin, L. C. Kwek, V. Vedral arxiv: (submitted to PRX, almost accepted)

2 Entanglement Entanglement: fundamental quantum property Different reasons for interest: 1. Quantum information quantum computers 2. Quantum Phase Transitions universality 3. Condensed matter non-local correlator 4. Integrable Models new playground 5. Cosmology Black Holes 6. Entanglement and Quantum Computation in Ising Chain n. 2 Fabio Franchini

3 Entanglement: what is it good for? Characterization of quantum states and how to simulate them (DMRG, MPS..) Detection of novel quantum phases (topological phases) Can determine computational power of a quantum phase? Does a quantum phase transition change such comp. power? Our answer: if QPT yields degeneracy from edge states the long-range order of these boundary states gives phase a greater quantum computational power Entanglement and Quantum Computation in Ising Chain n. 3 Fabio Franchini

4 Understanding Entanglement Consider a unique (pure) ground state Divide system into two Subsystems: A & B If system wave-function: jª A;B i = jª A i jª B i No Entanglement jª A;B i = DX p jjª A j i jªb j i j=1 Entangled jª A j i jªb j i (with D > 1, & linearly independent): Entangled: Measurements on B affect A Entanglement and Quantum Computation in Ising Chain n. 4 Fabio Franchini

5 Entanglement and Quantum Computation in Ising Chain n. 5 Fabio Franchini Von Neumann & Renyi Entropies jª A;B i = dx p jjª A j i jª B j i j=1 ½ A = tr B jª A;B ihª A;B j = X jjª A j i hª A j j Von Neumann (Quantum analog of Shannon Entropy): S A = tr A (½ A log ½ A ) = X j log j Renyi Entropy Entanglement spectrum S = 1 1 log tr (½ A) = 1 1 log X j j (equal to Von Neumann for α 1) Remark: S B = tr B (½ B log ½ B ) = S A

6 LOCC & Entanglement Consider bi-partite states (A B ): jª A;B i & j A;B i Entanglement cannot increase under Local Operations & Classical Communications (LOCC) if S ([ ]) < S ([ª]) 8 jª A;B i can be converted to (Depends upon partition choice!) j A;B i but not vice-versa! S. Turgut JPA (2007) A state can only be converted to one of lower entanglement Entanglement and Quantum Computation in Ising Chain n. 6 Fabio Franchini

7 Local Convertibility Take two bipartite states: jª A;B i & j A;B i If 9 1 such that S 1 ([ ]) < S 1 ([ª]) & 9 2 such that S 2 ([ ]) > S 2 ([ª]) the two states cannot be transferred locally (by LOCC) one into the other Entanglement and Quantum Computation in Ising Chain n. 7 Fabio Franchini

8 Local Convertibility & Adiabatic Evolution Adiabatic evolution: jª A;B i ground state of H(g) and j A;B i ground state of H(g + g) S α (g) monotonous S α (g) non-monotonous Study Renyi entropy derivative w.r.t g as function of α Differential Local Convertibility Entanglement and Quantum Computation in Ising Chain n. 8 Fabio Franchini

9 Local Convertibility & Entropy derivative Adiabatic evolution: Renyi entropy of instantaneous ground state of Hamiltonian H(g) as function of g and α If ds dg changes sign as α varies LOCC cannot simulate evolution Sign of entropy derivative distinguishes computational power of different phases Entanglement and Quantum Computation in Ising Chain n. 9 Fabio Franchini

10 Field Theory / Universality Naively, we expect all entanglement entropies to increase with the correlation length jª A;B i = dx p jjª A j i jª B j i j=1 S = 1 1 log tr (½ A) = 1 1 log X j Approaching a QPT, scale invariance require more eigenvalues to contribute equaly: DX Tr½ A = j = 1 ;! j ' 1 D j=1 ½ A = X jjª A j i hª A j j Entanglement and Quantum Computation in Ising Chain n. 10 Fabio Franchini j

11 Entanglement and Quantum Computation in Ising Chain n. 11 Fabio Franchini

12 Cui et al. Nature Comm. (2012) Numerical Results H I = NX j=1 ³ ¾ x j ¾x j+1 + g ¾z j g S Ising model for N=12 and bipartitions (6 6), (7 5), (8 4) Sign of entropy derivative: Blue = Negative; Red = Positive Ferromagnetic phase more powerful for adiabatic quantum computation! Not true for large subsystems! Entanglement and Quantum Computation in Ising Chain n. 12 Fabio Franchini

13 Local Convertibility & Topological Order Cluster Ising Model: (50 50) ( ) Entanglement and Quantum Computation in Ising Chain n. 13 Fabio Franchini

14 Local Convertibility & Topological Order The λ-d Model: (50 50) (96 4) Entanglement and Quantum Computation in Ising Chain n. 14 Fabio Franchini

15 Local Convertibility & Topological Order Perturbed 2-D Toric Code: Entanglement and Quantum Computation in Ising Chain n. 15 Fabio Franchini

16 The Quantum Ising Chain H I = NX j=1 ³ t ¾ x j ¾x j+1 + h ¾z j Exact (non-local) mapping into free fermions (Jordan-Wigner + Bogolioubov rotation) H I = X q µ " q  y q  q 1 2 ; " q = p t 2 + h 2 2ht cos q h=t > 1! h¾ x i = 0 h=t < 1! h¾ x i 6= 0 h=t = 1 Paramagnetic phase Ferromagnetic phase Ising QPT: c=1/2 Entanglement and Quantum Computation in Ising Chain n. 16 Fabio Franchini

17 H I = NX j=1 Kitaev Chain ³ t ¾ x j ¾x j+1 + h ¾z j = NX j=1 ³ t f (2) j f (1) j+1 + h f (1) j f (2) j Kitaev (2001) h=t! 1 Majorana Fermion f (1) j Majorana Fermion f (2) j ¾ x j ¾ y j Q l<j ¾z l Q l<j ¾z l c j = f (1) j + if (2) j h t h=t! 0 Edge State c j = f (1) j+1 + if (2) j Edge State Entanglement and Quantum Computation in Ising Chain n. 17 Fabio Franchini

18 H I = NX ³ t ¾ x j ¾x j+1 + h ¾z j Edge States = NX j=1 j=1 ³ t f (2) j f (1) j+1 + h f (1) j f (2) j t h t h t h t h t h t h t > 1 subsystem B subsystem A subsystem B h t < 1 subsystem B subsystem A subsystem B Entanglement and Quantum Computation in Ising Chain n. 18 Fabio Franchini

19 Edge States Entanglement Edge states combined into a complex fermion: occupied/empty two-fold degeneracy Long-range entanglement among edge states Edge states also generated by partitioning Grow closer as correlation length increases h t < 1 subsystem B subsystem A subsystem B Entanglement and Quantum Computation in Ising Chain n. 19 Fabio Franchini

20 EPR Analogy S = 0 S = ln 2 j0i j "#i j #"i 0 < S < ln 2 Approaching the QPT, edge states effectively grow closer their entanglement can decrease (while bulk states entanglement increases) Entanglement and Quantum Computation in Ising Chain n. 20 Fabio Franchini

21 Conclusions Entanglement derivative to study non-local convertibility Local way of detecting long-range entanglement! Edge state recombination explains it Approaching a QPT: 1. Correlation length increases 2. Bulk states entanglement increases 3. Edge states entanglement decreases Universal quantum simulator cannot be locally convertible Thank you! Entanglement and Quantum Computation in Ising Chain n. 21 Fabio Franchini

22 Quantum Computers & Simulators Certain problems too complex for classical computers: factorization, searches, simulation of quantum systems Quantum algorithms give exponential speed-up, but implementation of quantum computers is hard Quantum systems as computers Universal quantum simulator Entanglement and Quantum Computation in Ising Chain n. 22 Fabio Franchini

23 Quantum Adiabatic Algorithm Ground state of H I is the output of given problem Start from ground state of easy Hamiltonian H 0 Adiabatically evolved it to desire state H(t) = µ 1 t T H 0 + t T H I If velocity sufficiently small ( T 2 min stays in instantaneous ground state ), system Entanglement and Quantum Computation in Ising Chain n. 23 Fabio Franchini

24 Computational power Any efficient quantum algorithm can be casted as a Quantum Adiabatic Algorithm Adiabatic evolution performs quantum computation computational power of a quantum phase How to extract this computational power Entanglement! Entanglement and Quantum Computation in Ising Chain n. 24 Fabio Franchini

25 Entropy as a measure of entanglement Assume Bell State as unity of Entanglement: jbelli = j # #i j " "i p 2 ; j # "i j # "i p 2 Von Neumann Entropy measures how many Bell-Pairs can be distilled using LOCC from a given state jª A;B i (i.e. closeness of state to maximally entangled one) What can entanglement entropy teach us about a system? Entanglement and Quantum Computation in Ising Chain n. 25 Fabio Franchini

26 Z 2 Symmetry H I = NX j=1 ³ t ¾ x j ¾x j+1 + h ¾z j Ising model: prototype of symmetry Realized non-locally: string order parameter: Eigenstates with Z 2 Z 2 symmetry: h¾ x i = 0 ¹ x N = NY j=1 ¾ z j thermal ground state Symmetry broken states: h¾ x i 6= 0 Entanglement and Quantum Computation in Ising Chain n. 26 Fabio Franchini

27 j0i = 2 L X =1 Entanglement p jª A i jª B i S = 1 1 log X Quadratic Theory: Block eigenstates from block excitations jª A i = jn 1 ; n 2 ; : : : ; n L i ; n l = 0; 1 = jhª A j0ij2 = LY h0jn j ihn j j0i j=1 Measure overlap of block excitaions with G.S.: Whole system excitations: Block excitation: c j ; c y j! c jj0i = 0 ~c l ; ~c y l! ~c jj0i 6= 0 Entanglement and Quantum Computation in Ising Chain n. 27 Fabio Franchini

28 Entanglement j0i = 2 L X =1 p jª A i jª B i S = 1 1 log X Block excitations from correaltion matrix: Vidal & al, PRL (2003) hf (a) k f (b) j i = ± j;k ± a;b + i (B L ) (a;b) (j;k) eigenvalues iº j h0j0 j ih0 j j0i = h0j~c j ~c y j j0i = 1 + º j 2 h0j1 j ih1 j j0i = h0j~c y j ~c jj0i = 1 º j 2 Overlap between block excitations and ground state = LY h0jn j ihn j j0i = j=1 LY µ 1 ºj j=1 2 Entanglement and Quantum Computation in Ising Chain n. 28 Fabio Franchini

29 Correlation Matrix Eigenvalues hf (a) k f (b) j i = ± j;k ± a;b + i (B L ) (a;b) (j;k) h0jd j d y j j0i = 1 + º j 2 h0jd y j d jj0i = 1 º j 2 L=2 L=10 One edge state for h<1: partial overlap Approaching QPT: bulk states overlap decreases, edge states overlap increases (edge state recombination) Entanglement and Quantum Computation in Ising Chain n. 29 Fabio Franchini

30 2-Sites Block Entanglement Lack of local convertibility due to edge state recombination 2-sites classical gates destroy long-range correlations! Entanglement and Quantum Computation in Ising Chain n. 30 Fabio Franchini

31 Large Block Entanglement For L! 1 we have full analytical knowledge of entanglement (spectrum) h=t < 1 For edge states give double degeneracy Local convertibility restored! Numerics confirm Its & al. (2005); F.F. & al. (2008); F.F & al. (2011) Entanglement and Quantum Computation in Ising Chain n. 31 Fabio Franchini

32 Symmetry broken Ground State So far, ground state as eigenstate of Z 2 : ¹ x N = For h < 1 ; h¾ x i 6= 0 : symmetry broken state no edge states locally convertible! No analytical approaches, just numerics NY j=1 ¾ z j ( ) ( ) Entanglement and Quantum Computation in Ising Chain n. 32 Fabio Franchini

33 Conclusions Non-local convertibility from entanglement derivative Local way of detecting long-range entanglement! Edge state recombination explains it Approaching a QPT: 1. Correlation length increases 2. Bulk states entanglement increases 3. Edge states entanglement decreases Universal quantum simulator cannot be locally convertible Thank you! Entanglement and Quantum Computation in Ising Chain n. 33 Fabio Franchini

34 Entanglement Spectrum First few eigenvalues of the reduced density matrix (multiplicities not shown) Finite Size Numerical results Theromodynamic Limit Analytical results Entanglement and Quantum Computation in Ising Chain n. 34 Fabio Franchini

35 Renyi Entropy Entropy depends on single parameter ε ε vanishes at phase transitions, large in gapped phase Microscopics of the model through ε(k) (2) h¾ x i = 0 (1a) Ω + h > 1 : k = p h ! dk dh < 0 (1b) h¾ x i 6= 0 h < 1 : k = p h ! dk dh > 0 γ Entanglement and Quantum Computation in Ising Chain n. 35 Fabio Franchini

36 Entanglement Derivative Paramagnetic Phase h=2 h=1.1 dk dh < 0 h=.6 h=.9 Ferromagnetic Phase dk dh > 0 Entanglement and Quantum Computation in Ising Chain n. 36 Fabio Franchini

37 L-spins subsystem Diagonalize L x L Hankel matrix: 0 1 g L 1 g L 2 : : : g 0 g L 2 g L 3 : : : g 1 ~B = C A ; g j 1 2¼ g 0 g 1 : : : g 1 L Z 2¼ 0 e ijµ cos µ h + i sin µ q (cos µ h) sin 2 µ dµ Use L eigenvalues λ j to compute Renyi entropy as sum of entropies of 2-levels systems: S( ) = 1 1 ds( ) = 1 LX µ µ 1 + l 1 l ln l=1 LX l=1 (1 + l) 1 (1 l) 1 (1 + l) + (1 l) d l Entanglement and Quantum Computation in Ising Chain n. 37 Fabio Franchini

38 2-spins subsystem: Ising line Entropy derivative vanishes in ferromagnetic phase! the two phases have different computation power! Role of Majorana edge states? Entanglement and Quantum Computation in Ising Chain n. 38 Fabio Franchini