Power law violation of the area law in quantum spin chains

Transcription

1 Power law violation of the area law in quantum spin chains and Peter W. Shor Northeastern / M.I.T. QIP, Sydney, Jan. 2015

2 Quantifying entanglement: SVD (Schmidt Decomposition, aka SVD) Suppose ψ AB is the pure state of a composite system, AB. Then there exists orthonormal states Φ α A for A and orthonormal states θ α B for B ψ AB = λ α Φ α A θ B α α where λ α s satisfy α λ α 2 = 1 known as Schmidt numbers. The number of non-zero Schmidt numbers is called the Schmidt rank, χ, of the state (a quantification of entanglement).

3 Quantifying entanglement: Entropy Another measure of entanglement is entanglement entropy. Recall we had ψ AB = λ α Φ α A θ B α α where λ α s satisfy α λ α 2 = 1. The entanglement entropy is: S λ α 2 log λ α 2 α where λ α 2 = p α are the probabilities.

4 Area laws Area Laws nomena, one thinks less of very detailed properties, b rather interested in the scaling of the entanglement ent when the distinguished region grows in size. In fact, for q tum chains, this scaling of entanglement as genuine qua correlations a priori very different from the scaling of point correlation functions reflects to a large extent the ical behavior of the quantum many-body system, and sh some relationship to conformal charges. At first sight one might be tempted to think that the ent of a distinguished region I, will always possess an exten character. Such a behavior is referred to as a volume sc and is observed for thermal states. Intriguingly, for ty FIG. 1 A lattice L with a distinguished set I L (shaded area). Vertices depict the of I with surface area s(i) ground states, however, this is not at all what one encoun Instead, one typically finds an area law, or an area law a small (often logarithmic) correction: This means that i distinguishes a region, the scaling of the entropy is m of the region I, the state will not be pure in general and will have a non-vanishing von-neumann entropy S( I). 1 linear in the boundary area of the region. The entangle entropy is then said to fulfill an area law. It is the purpo In contrast to thermal states this entropy does not originate this article to review studies on area laws and the scalin from a lack of knowledge about the microstate of the system. Even at zero temperature we will encounter a non-zero The main four motivations to approach this que the entanglement entropy in a non-technical manner. entropy! This entropy arises because of a very fundamental (known to the authors) are as follows: property of quantum mechanics: Entanglement. This quite intriguing trait of quantum mechanics gives rise to correlations The holographic principle and black hole entr even in situations where the randomness cannot be traced back The historical motivation to study the entangleme to a mere lack of knowledge. The mentioned quantity, the entropy of a subregion is called entanglement entropy or geomet- hole physics: It has been suggested in the seminal geometric entropy stems from considerations of b ric entropy and, in quantum information, entropy of entanglement, which represents an operationally defined entanglement for a discrete version of a massless free scalar fie of refs. 23,207 that the area law of the geometric ent measure for pure states (for recent reviews see refs. 125,186 ). then numerically found for an imaginary sphere in In the context of quantum field theory, questions of scaling Ramis of entanglement Movassagh entropies in themovassagh size of I have some andtra- Shor: arxiv: holes, 118 in particular[quant-ph] the Bekenstein-Hawking dial symmetry could be related to the physics of b ent Picture from Eisert, Cramer, Plenio, Rev. Mod. Phys. 82 (2010) Area law: Suppose you have a Hamiltonian with only local interactions, and a quantum system is in the ground state of the Hamiltonian. Then the entropy of entanglement between two subsystems of a quantum system is proportional to the area of the boundary between them.

5 Area law and implications for simulability 1D gapped local systems obey an area law. [M.B. Hastings (2007)] This makes them easy to simulate on a classical comuter. Matrix Product States, DMRG, PEP, etc. work very well for 1D systems with an area law.

6 higher dimensions It is believed that higher-dimensional gapped systems obey an area law (open). For critical systems, it is believed the area law contains an extra log factor. In D spatial dimensions one expects: S L D 1 : Gapped S L D 1 log(l) : Critical

7 Phase transitions For 1-dimensional spin chains at critical points, the continuous limit is generally a conformal field theory: Entropy of entanglement: O(log n), Spectral gap: O(1/n).

8 Basic idea that started our research Simulating 1D spin chains with local Hamiltonians is BQP-complete. (Gottesman, Irani). 1D spin chains with low entanglement are classically simulable. Therefore: there must be 1D spin chains with high entanglement.

9 arxiv: Movassagh, Farhi, Goldstone, Nagaj, Osborne, Shor (2010) We investigated spin chains with qudits of dimenision d, interaction is a projection dimension r. The ground state is frustration-free but entangled when d r d 2 /4. we could compute the Schmidt ranks, We could not obtain definitive results on the spectral gap or the entanglement entropy.

10 arxiv: Irani (2010) There are Hamiltonians whose ground states have: spectral gap O(1/n c ), entanglement entropy O(n), complicated Hamiltonians, high-dimensional spins.

11 Bravyi et al 2012 Bravyi, Caha, Movassagh, Nagaj, Shor (2012) There are Hamiltonians whose ground states have spectral gap O(1/n c ), have entanglement entropy O(log n), are frustration free, have spins of dimension 3.

12 New result Movassagh, Shor (2014) There are Hamiltonians whose ground states have spectral gap O(1/n c ), c 2 have entanglement entropy O( n), are frustration free, have spins of dimension 2s + 1, s > 1.

13 Another new result Movassagh, Shor (2014) There are Hamiltonians whose ground states numerically have spectral gap O(1/n c ), c 2 have entanglement entropy O( n), are unique, are not frustration free, have spins of dimension 2s + 1, s > 1. These properties do not depend on the boundary conditions.

14 Summary of the new result The Motzkin state, M 2n,s is the unique ground state of the local Hamiltonian Entanglement entropy violates the area law: S (n) = c 1 (s)log 2 (s) n log(n) + c 2 (s) χ = sn+1 1 s 1. The gap upper bound: O ( n 2). Brownian excursion and universality of Brownian motion. The gap lower bound: Ω((n c ), c 1. Fractional matching and statistics of random walks

15 States: d = 2s + 1 s = 1 l 1 ( 0 0 d = 3 r 1 )

16 States s = 1 l 1 ( 0 0 d = 3 r 1 )

17 s 1 s = 2 l 1 l 2 0 r 1 r 2 ( [ 0 ) ] d = 5

18 Ground states

19 How to quantify entanglement Entanglement of Motzkin States is due to the mutual information between halves Motzkin walk A B ( ) ( 0 ( ( 0 ) ) )

20 How to quantify entanglement Entanglement of Motzkin States is due to the mutual information between halves Motzkin walk m A B ( ) ( 0 ( ( 0 ) ) )

21 More than one type of parenthesis e.g., s = 2 Suppose there are two types ( and { to match { } ( 0 { ( 0 ) } )

22 More than one type of parenthesis e.g., s = 2 Entanglement of Colored Motzkin States is due to the mutual information between halves Motzkin walk { } ( 0 { ( 0 ) } )

23 Subtlety for s > 1 Suppose there are two types ( and { to match ( 0 { 0 } ) O.K.

24 Matching that does NOT work Suppose there are two types ( and { to match ( 0 { 0 ) } Not O.K.! ( 0 { 0 } ) O.K.

25 The ground state s = 1 e.g., 2n = {2,4} M 2n,s = 1 M2n p th Motzkin walk p M 2 = 1 { 00 + lr } 2 M 4 = 1 { lr + 0l0r + l00r + 0l0r 9 + l0r0 + lr00 + lrlr + llrr }

26 The ground state s 1 e.g., 2n = 2 M 2n,s = 1 M2n p th s-colored Motzkin walk M 2,s p { 00 + s k=1 l k r k }

27 Entanglement: Schmidt rank χ p n,m,s = M2 n,m,s, N n,s N n,s n m=0 s m M 2 n,m,s, (1) Geometric sum on m gives χ = sn+1 1 s 1 and entanglement entropy is S ({p n,m,s }) = n m=0 s m p n,m,s log 2 p n,m,s. (2)

28 Combinatorial factors M n,m,s : number of walks starting at zero ending at height m with s total colors M n,m,s = m + 1 ( n + 1 i 0 M n,m,i,s i 0 n + 1 i + m + 1 i n 2i m ) s i

29 Quantifying entanglement, area laws and simulability Location of the Saddle # M n,m,i,s, n = 90 and s = i Saddle point: Mn,m,s,i +1 Mn,m,s,i = 1, m 0 Mn,m+1,s,i Mn,m,s,i = 1.

30 Saddle point approximation Turning the sum into an integral (carefully) and performing saddle point integration m = α n : M n,m,s S 2log(s) σ ( 1 s 2 πσn 3/2 σ 2σ π s s. ) n+1 ) αs α n/2 exp( α2 4σ n logn + γ log(2πσ),.

31 Underlying Hamiltonian implements local moves H = { s k=1 r k 1 r k + s k=1 } 2n 1 l k 2n l k + Π j,j+1 projects onto the span of ( k,= 1,,s) 1 ] [0l k l k ] [0r k r k ] [00 l k r k 2 j=1 : 0l k l k 0 : 0r k r k 0 : 00 l k r k Π cross = l k r i r i l k. k i Π j,j+1,

32 Meaning of terms in Hamiltonian The terms 1 ] [0l k l k ] [0r k r k ] [00 l k r k 2 : 0l k l k 0 : 0r k r k 0 : 00 l k r k enforce an equal superposition of all walks which can be reached by switching l k and 0 switching r k and 0, replacing l k r k by 00.

33 Meaning of terms in Hamiltonian The cross terms Π cross = l k r i r i l k. k i ensure that the types of parentheses match. The boundary terms { s k=1 r k 1 r k + ensure that the walk is balanced. s k=1 l k 2n l k }

34 Gap Upper-Bound We show that the gap is O(n 2 ).

35 Gap: Upper bound I We want any state φ such that φ H φ = O ( n 2), φ ground H φ < 1 2. Then φ = α g φ g + α 1 φ 1 + α 2 φ and φ H φ = α 2 1 φ 1 H φ 1 + α 2 2 φ 2 H φ 2 + α 2 3 φ 3 H φ (1 α 2 g ) φ 1 H φ 1

36 Gap: Upper bound II φ = 1 M2n e p { 2πiθ ( Area of p th )} Motzkin walk p th Motzkin walk M 2n φ = 1 M 2n e p { 2πiθ ( Area of p th )} Motzkin walk

37 Gap: Upper bound II φ = 1 M2n e p { 2πiθ ( Area of p th )} Motzkin walk p th Motzkin walk M 2n φ = 1 M 2n e p { 2πiθ ( Area of p th )} Motzkin walk lim 2n φ F A (θ) n 0 f A (x)e 2πixθ dx.

38 Gap: Upper bound III f A (x) F A(θ) x f A (x) = 2 6 x 2 j=1 ( v 2/3 j e v j U 5 6, 4 ) 3 ;v j θ x [0, ) v j = 2 a j 3 /27x 2 where a j are the zeros of the Airy function.

39 Gap Lower-Bound Θ(n c ), for some constant c. We use the same techniques as in Bravyi, Caha, Movassagh, Nagaj, Shor. the projection lemma relating Motzkin walks and Dyck walks, proving rapid mixing with the canonical paths method, fractional matchings.

40 This Hamiltonian isn t completely satisfactory requires boundary conditions to have unique ground state. Without the boundary conditions, there would be ( ) n+2 2 ground states, each coming from a superposition of unbalanced walks: ( 0 ( ( ) 0 ) () How can we eliminate these ground states without boundary conditions? We add an energy penalty for l and r i.e., for ( and ).

41 Hamiltonian with extermal magnetic field. How can we prove anything about these states? The argument that the gap is at most O(n 2 ) still holds. Only have to worry about the gap in two cases: Unbalanced walks Superposition of balanced walks with positive coefficients.

42 Hamiltonian with extermal magnetic field. The gap for unbalanced walks: Let ε be the energy penalty for l k, r k. We can use perturbation theory (backed up with numerics) to show that these ground states have an increased energy of around cε 2 /n.

43 Hamiltonian with extermal magnetic field. The gap for states in the balanced subspace. There is a polynomial gap in the balanced subspace in the Hamiltonian without an energy penalty. It appears from numerics (on chains of relatively small length) that the gap in this case is Numerics seem to show that the gap in the balanced subspace with an energy penalty is also Θ(n 2 ).

44 Open problems Is there a continuum limit for these Hamiltonians? Can we rigorously prove the results with an external magnetic field? Are there frustration-free Hamiltonians with unique ground states which violate the area law by large factors?

45 Lastly... Thank you