Many-body entanglement witness

Transcription

1 Many-body entanglement witness Jeongwan Haah, MIT 21 January 2015 Coogee, Australia arxiv:

2 Quiz Energy Entanglement Charge Invariant Momentum Total spin Rest Mass Complexity Class

3 Many-body Entanglement Local entanglement can be washed away by local unitaries. Equivalence relation among states: = Transitivity: If A=B and B=C, then A=C i Many-body Entanglement is an equivalence class under Small-depth Quantum Circuits

4 Many-body Entanglement Topological order is long-range entanglement pattern. Topological order is the coarsest structure of the state. Should be easy to detect How would we recognize the pattern?

5 Guiding Problem How deep a quantum circuit must be in order to transform a state to another? Can an invariant answer this question by a significant bound? Strength or fineness (opp. coarseness) of the invariant.

6 0. Long-range order

7 Quantum circuits V = V It takes a linear depth-circuit to build up any long-range correlation. O 1 O 2,Cor W i (O 1,O 2 ) 0 Cor i (O 1,O 2 ) 0

8 Finite correlation length Long-range Entanglement? Long-range correlation Many exactly solvable models have commuting Hamiltonian Quantum double models, Levin-Wen model, any Pauli stabilizer code state. AB A B =0 NOT TOO GOOD

9 0. Long-range order 1. Local Indistinguishability

10 Hardness of Generation [Wolfram MathWorld] Bravyi, Hastings, Verstraete (2006) If= 0 i 0 i 1 i The pair is locally distinguishable. Deep Entanglement i Any orthogonal state is locally distinguishable. The local indistinguishability is invariant of a pair of states. A locally indistinguishable partner is an entanglement witness.

11 Toric code on a sphere No correlation of local observables. No pair of locally indistinguishable states. H = X e2 z e X e3v x e NOT TOO GOOD What is the complexity of generation? Is there deep entanglement?

12 0. Long-range order 1. Local Indistinguishability 2. Topological Entanglement Entropy

13 Topological Entanglement Entropy Kitaev, Preskill; Levin, Wen (2006) S A = L A = log s X d 2 a a total quantum dimension Kitaev-Preskill Argument A B S A + S B + S C S AB S BC S CA + S ABC C

14 Topological Entanglement Entropy S A = L (Simply) Computable in the bulk Quantitative Many-body entanglement witness Connected to abstract anyon theory Specific to 2D

15 AntiFerroHeisenberg on Kagome Yan, Huse, White (2011) Found no ordering under perturbations Jiang, Wang, Balents (2012) Computed topological entanglement entropy Strong evidence of topological order.

16 Bravyi s Counterexample From his talk in 2008 H = X i X i QC of depth 2 A B H = X i Z i 1 X i Z i+1 C S even = L/2 1 S Kitaev-Preskill = log 2

17 2D cluster state on triangular lattice S = L - 1 [Zou, Haah, Senthil, in preparation] Sub-leading term of E.Entropy can be contaminated. It can even fluctuate. S(L) =L gcd(l, n) NOT TOO GOOD Consequence of 1D SPT under a product group Can we say that TEE is an evidence for topological order?

18 0. Long-range order 1. Local Indistinguishability 2. Topological Entanglement Entropy 3. Small-depth stabilizers

19 Small-depth Stabilizers They are locally invisible. Z Z Z Z Z Z Z Z looks the same Did you apply it? looks the same Z Z Z Z Z Z Z Z

20 Locally invisible operator A B Def.: O is (A,B)-locally invisible with respect to i Tr B c[ ih ] =Tr B c[ ih )Tr A c[o ih O ] / Tr A c[ ih ] Small-depth stabilizing quantum circuit is (A,A+r)-locally invisible.

21 Twist product Ordinary product PQ Ordinary product QP P Q P Q Twist Product P Q X ij P (i) up Q (j) up Q (j) down P (i) down Well-defined as long as intersection is separated.

22 For product states P Q h P 1Q i = h P ih Q i Any pair of locally invisible operators whose twist pairing is nontrivial, is a witness of deep entanglement.

23 Examples Z X Optimal bound on generating circuits! Z X Far-separated Bell pair H = X e2 z e X e3v x e Toric Code state

24 Witness, nice!

25 0. Long-range order 1. Local Indistinguishability 2. Topological Entanglement Entropy 3. Small-depth stabilizers 4. Topological Charges

26 Topological S-matrix Quantum amplitude of braiding process D 2 = X a d 2 a S ab = 1 D h a i = d a a b Invariant of Hamiltonian or state?

27 Minimally Entangled States Zhang, Grover, Oshikawa, Vishwanath (2012) Zhang, Grover, Vishwanath ( ) Start with full ground space. Compute minimal ent. states. Compute overlap. S ab = h H a V b i Can we do it in the bulk?

28 Goal Find a quantity such that It is defined by a state. It is independent of boundary conditions. It is invariant under local unitary transformations. (It can be computed given a wave function.)

29 What is anyon? It is a superselection sector. A set of states related by local operators, not necessarily unitary. No symmetry constraint. Looks identical to ground state. Arbitrary operator Irrelevant to define particle type in the disk

30 Recall: Total spin [J x,j y ]=ij z J 2 x + J 2 y + J 2 z = j(j + 1) Allowed operators, Find an operator in the center of the operator algebra. Eigenvalue of the central operator = Particle type (spin) = Conservation

31 To define particle types Mat(D) A Any local term of H should commute Looks identical to ground state. Arbitrary operator Allowed operators, Find an operator in the center of the operator algebra. Eigenvalue of the central operator = particle type (spin)

32 Null operators Looks identical to ground state. Arbitrary operator If any operator on grey annihilates the state, it s like multiplying by 0. Factor them out. Mat(D) A/N Operator on grey that annihilates the state Any local term of H should commute

33 C*-algebra Algebra over complex numbers (finite dimensional) Enough to think of matrix algebra closed under dagger. Completely decomposes into (a direct sum of) full matrix algebras Projections onto components generate the center.

34 Structure of C*-algebra 2 3 UCU = = I , 2 = I 4. 8 >< = I j >: 2 = j = j 1 2 =0

35 Particle type projectors form the canonical basis of the center of Mat(D) A/N The center lives on the annulus. Looks identical to ground state. Structure theorem of C*-algebra Arbitrary operator

36 My S-matrix S PQ =h i P Q Particle type projectors Input: (commuting) Hamiltonian (ground state) No special boundary; just some large disk. No phase ambiguity. The trivial particle ( 1 ) projector is distinguished.

37 Relation to ord. S-matrix Proof: S ab = d ad b D S ab It contains the same data! a = d a D X b S ab b

38 Invariance under local unitaries h W W W W i P Q Particle type projectors V W (P 1Q)W =(WPW )1(WQW ) as long as the depth of W is smaller than the separation of the intersection. So, invariance is proved if A/N is remains isomorphic under W. This is nontrivial, so I had to assume further.

39 Assumptions 1. Local topological order Local ground state matches the global one 2. Stable logical algebra logical algebra does not depend on the size of the support violated when there are infinitely many particle types.

40 Local Topological Order A

41 Stable Logical Algebra Isomorphic A/N Regardless of the thickness

42 Finiteness of particle types Infinite stack of 2D layers A particle is separated by a sphere with thick wall. Side View Stable logical algebra is nontrivial assumption in general.

43 Consequences A/N is in fact independent of Hamiltonian is invariant under small-depth Q. circuit. Therefore, my S-matrix is an invariant of state. UHU H H 0 U i i S

44 Complexity of transformation Any transformation between states with distinct S- matrices requires a deep (linear in diameter) circuit. H 0 UH 0 U 6= H 1 0 i U 0 i = 1 i In view of quasi-adiabatic evolution, the energy gap must close at some point in any path between Hamiltonians with distinct S-matrices.

45 Sketch of independence proof A t /N t!i t /M t!a t+w /N t+w Logical algebra to locally invisible operators Locally invisible operators to logical algebra They are naturally invisible thanks to local topological order condition. Symmetrize so locally invisible operators is dressed to commute with the Hamiltonian A H 1 t /N H 1 t!i t /M t!a H 2 t+w/n H 2 t+w

46 Toric code state Abelian discrete gauge theory A/N is diagonal matrix algebra of dimension d2 S (d) (a x a z ),(a 0 x a0 z ) = 1 d 2!a za 0 x +a xa 0 z d. Two assumptions are satisfied, as verified by direct computation. Rows and columns unsorted except for the distinguished 1. Verlinde formula recovers the fusion (group) rules.

47 Row-column matching P Q If projectors jointly stabilize some state, they are matched.

48 0. Long-range order 1. Local Indistinguishability 2. Topological Entanglement Entropy 3. Small-depth stabilizers 4. Topological Charges

49 Many-body Entanglement Witness S PQ =h i P Q We have given a class of ground states, for which S-matrix can be defined. Only a patch of a ground state is needed; insensitive to boundary. Indeed invariant under perturbations. 2D is not particularly used. Any heuristic algorithm would be interesting. Perhaps, in 2D stable logical algebra assumption is redundant.