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Entanglement and Quantum Spin Chains

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Entanglement and Quantum Spin Chains ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems ENTANGLEMENT AND QUANTUM SPIN CHAINS V. Korepin YITP of Stony Brook University New exactly solvable spin chain came out of quantum information V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Outline 1 ABitofHistoryofQuantumEntanglement 2 Quantum Information and Dynamical Systems V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems The dawn Aug 14, 1935 Shroedinger submitted his paper to Proceedings of Cambridge Philosophical Society. 1964 John Bell: entanglement distinguishes quantum mechanics from classical at fixed value of Plank constant. 1981 Richard Feynman: quantum computation. 1996 Bennett, Bernstein, Popescu and Schumacher introduced entropy of subsystem as a measure of entanglement discovery of entanglement entropy V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Application to physics: area Law 1986 L. Bombelli, R. Koul, J. Lee, R. Sorkin; 1993 M. Srednicki 1994 Holzhey, Larsen and Wilczek: ent entr of block of spins of c size x for 1D gapless models scales as S = 3 log x . 2003 Logarithmic scaling of Renyi entropy. B.-Q.Jin, Korepin, Journal Statistical Physics; arXiv.org April 15 of 2003. ln tr(ρα) −1 S (α)= 1+α ln x R 1−α → 6 c v πTx 2004 For finite temperature S = 3 log πT sinh v Korepin, PRL, vol 92, ei 096402, 05 March 2004,! ! "" arXiv:cond-mat/0311056 V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Environment and Error Correcting Codes Q-bits should be entangled with other qubits in quantum computer, but not entangled with the environment. Error correcting codes discovered by Peter Shor 1995. Gottesman developed Stabilizer Codes 1997. Eˆn Ψ = Ψ | ⟩ | ⟩ Set of Eˆn are error operators: bit flip, phase flip... Linear combination of all Ψ is a error-free subspace. Sometimes Eˆn | ⟩ can be expressed in terms of local spins and Eˆn 1. For | | ≤ H = Eˆn error-free subspace is the ground state. The − n most famous! example is KITAEV MODEL. Recently Peter Shor designed another solvable model of statistical mechanics starting from error correcting codes. V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems New spin chain New spin chain come out of error correcting codes. Ramis Movassagh and Peter Shor: arXiv:1408.1657 The Hamiltonian is translation invariant. It describes interaction of nearest neighbors in 1 dimension: H = Hj,j+1 "j Can be consturcted for any integer spins.. The ground state is unique and know explicitly. It is described in terms of enumerative combinatorics [Motzkin numers] . In the infinite volume L the model is gapless. The energy gap decays as L−∆ and ∆ 2. ≥ V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Combinatoric and Spin Chains Spin 2. In r.h.s I wrote the value of Sz l1 = 2 , l2 = 1 , r 1 = 1 , r 2 = 2 .Statesontwo lattice| ⟩ | sites:⟩ | ⟩ | ⟩ | ⟩ |− ⟩ | ⟩ |− ⟩ 1 Rk = 0, r k r k , 0 (1) | ⟩ √2 #| ⟩−| ⟩$ 1 Lk = 0, lk lk , 0 (2) | ⟩ √2 #| ⟩−| ⟩$ 1 ϕk = 0, 0 lk , r k (3) | ⟩ √2 #| ⟩−| ⟩$ k = 1, 2 (4) V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems The Hamiltonian 2n−1 cr H = ⎛ Πj,j+1 + Πj,j+1⎞ + Πboundary "j=1 ⎝ ⎠ 2 k k k k k k Π , = R R + L L + ϕ ϕ j j+1 | ⟩j ⟨ |j+1 | ⟩j ⟨ |j+1 | ⟩j ⟨ |j+1 "k=1 Πcr = lk r i lk r i j,j+1 | ⟩j ⟨ |j+1 "k≠ i 2 Π = r k r k + lk lk boundary | ⟩1⟨ |1 | ⟩2n⟨ |2n "k=1 V. Ko r e p in F-H Formula 2 m [ ] ( 0 [ ( 0 ) ] ) Figure 2: A Motzkin walk with s =2colors of length 2n = 10. Local projectors as interactions have the advantage of being robust against certain perturbations. This is important from a practical point of view and experimental realizations. The local Hamiltonian, with projectors as interactions, that has the Motzkin state as its unique zero energy ground state is 2n 1 2n 1 − − cross H = ⇧j,j+1 + ⇧j,j+1, (1) Xj=1 Xj=1 where ⇧j,j+1 implements the local changes discussed above and is defined by s ⇧ Rk Rk + Lk Lk + 'k 'k j,j+1 ⌘ ij,j+1h ij,j+1h ij,j+1h Xk=1 h i with Rk = 1 0rk rk0 , Lk = 1 0`k `k0 and 'k = 1 00 `krk . The i p2 | i| i i p2 | i i i p2 i i s k k k k projectors ⇧boundary⇥ k=1 ⇤r 1 r + ` 2⇥n ` select⇤ out the Motzkin state⇥ by excluding⇤ all ⌘ | i h | | i h | cross k i k i walks that start and end at non-zero heights. Lastly, ⇧j,j+1 = k=i ` r j,j+1 ` r ensures that P ⇥ ⇤ 6 | i h | balancing is well ordered (i.e., prohibits 00 `kri when k = i); these projectors are required only $ 6 P when s>1 and do not appear in the s =1model (PRL 109, 207202 (2012)). Lastly for the Hamiltonian to : 1. Be local (nearest neighbors), which H defined above is, 2. Translationally invariant, which H is, and 3. Have a unique ground state, which H has not, that violates the area law by log (n) for s =1 and pn for s =2, 3,... , we add an external field, where the model is described by the new Hamiltonian H˜ H + ✏ F ⌘ 2n s F [ r i r + ` i ` ] ⌘ ( | i h | | i h | ) Xi=1 Xk=1 where, H is as before and 0 < ✏ 1. H˜ has all the features above and has a unique ground ⌧ state. The gap ∆ = ⇥ (n c), where provably c is a constant c 2, which cannot have CFT as its − ≥ continuum limit. Notes on the spin-operator representation of the model 1, 2, Ramis Movassagh ⇤ 1Department of Mathematics, Northeastern University, Boston MA 02115 2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139 (Dated: May 19, 2015) We will write the spin interactions in the spin-representation for the model that gives a square root violation of the area law (joint work with Peter W. Shor arXiv:1408.1657 [quant-ph]). A. Review of the local Hamiltonian [arXiv:1408.1657 [quant-ph]] Let us consider an integer spin s chain of length 2n. It is convenient to label the d =2s +1 − spin states by 0, `1, `2, , `s,r1,r2, ,rs where ` means a left parenthesis (or a step up) and ··· ··· r a right parenthesis (or a step down) as shown in Fig. 1. We distinguish each type of steps (or parenthesis) by associating a color from the s colors shown as superscripts on ` and r. A Motzkin walk on 2n steps is any walk from (x, y)=(0, 0) to (x, y)=(2n, 0) with steps (1, 0), (1, 1) and (1, 1) that never passes below the x-axis, i.e., y 0. An example of such a walk is shown − ≥ in Fig. 2. The height in the middle is 0 m n which results from m steps up with the balancing steps down on the second half of the chain. In our model the unique ground state is the s colored − Motzkin state which is defined to be the uniform superposition of all s colorings of Motzkin walks on 2n steps. The nonzero heights in the middle is the source of the mutual information between the two halves and the large entanglement entropy of the half-chain (Fig. 2). Consider the following local operations to any Motzkin walk: interchanging zero with a non-flat step (i.e., 0rk rk0 or 0`k `k0) or interchanging a consecutive pair of zeros with a peak of a $ $ given color (i.e., 00 `krk). Any s colored Motzkin walk can be obtained from another one by a $ − sequence of these local changes. To construct a local Hamiltonian with projectors as interactions that has the uniform superposition of the Motzkin walks as its zero energy ground state, each of the local terms of the Hamiltonian has to annihilate states that are symmetric under these interchanges. s = 2 ℓ1 ( ℓ2 [ 0 0 d = 5 r1 ) r2 ] Figure 1: Labeling the states for s =2. 3 B. Hamiltonian in Spin-Operator language Recall that in the spin operator representation, taking ~ =1: S2 s, m = s (s + 1) s, m | i | i Sz s, m = m s, m | i | i S± s, m = s (s + 1) m (m 1) s, m 1 | i − ± | ± i p where S = sx isy. ± ± For a general spin s model, let k =1, 2,...s.Wehave − 1 Lk Lk = ( 0,k k, 0 )( 0,k k, 0 ) | ih | 2 { | i| i h | h | } 1 = ( 0 0 k k k 0 0 k 0 k k 0 + k k 0 0 ) , 2 { | ih | ⌦ | ih | − | ih | ⌦ | ih | − | ih | ⌦ | ih | | ih | ⌦ | ih | } 1 Rk Rk = ( 0, k k, 0 )( 0, k k, 0 ) | ih | 2 { | − i| − i h − | h− | } 1 = ( 0 0 k k k 0 0 k 0 k k 0 + k k 0 0 ) 2 { | ih | ⌦ | − ih | − | − ih | ⌦ | ih | − | ih | ⌦ | − ih | | − ih | ⌦ | ih | } 1 'k 'k = ( 0, 0 k, k )( 0, 0 k, k ) | ih | 2 { | i| − i h | h − | } 1 = ( 0 0 0 0 k 0 k 0 0 k 0 k + k k k k ) . 2 { | ih | ⌦ | ih | − | ih | ⌦ | − ih | − | ih | ⌦ | ih | | ih | ⌦ | − ih | } Moreover the ’cross terms’, such as k, j k, j where k = j, can simply be wrtten as k k j j .
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