Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap Power law violation of the area law in quantum spin chains Ramis Movassagh and Peter W. Shor Northeastern / M.I.T. QIP, Sydney, Jan. 2015 Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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 α B � 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) . Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap Quantifying entanglement: Entropy Another measure of entanglement is entanglement entropy . Recall we had | ψ AB � = ∑ λ α | Φ α A �⊗| θ α B � α where λ α ’s satisfy ∑ α | λ α | 2 = 1. The entanglement entropy is: | λ α | 2 log | λ α | 2 S ≡ − ∑ α where | λ α | 2 = p α are the probabilities. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap Area laws Area Laws 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. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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: L D − 1 ∼ : Gapped S L D − 1 log ( L ) : S ∼ Critical Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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 ) . Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap arxiv:1001.1006 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. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap arXiv:0901.1107 Irani (2010) There are Hamiltonians whose ground states have: spectral gap O ( 1 / n c ) , entanglement entropy O ( n ) , complicated Hamiltonians, high-dimensional spins. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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 2 s + 1, s > 1. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap 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 2 s + 1, s > 1. These properties do not depend on the boundary conditions. Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap Summary of the new result The ’Motzkin state’ , | M 2 n , s � is the unique ground state of the local Hamiltonian Entanglement entropy violates the area law: c 1 ( s ) log 2 ( s ) √ n + 1 S ( n ) = 2 log ( n )+ c 2 ( s ) s n + 1 − 1 = . χ s − 1 n − 2 � � The gap upper bound: O . Brownian excursion and universality of Brownian motion. The gap lower bound: Ω(( n − c ) , c ≫ 1. Fractional matching and statistics of random walks Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap States: d = 2 s + 1 s = 1 ℓ 1 ( 0 0 d = 3 r 1 ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap States s = 1 ℓ 1 ( 0 0 d = 3 r 1 ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap s ≥ 1 s = 2 ℓ 1 ( ℓ 2 [ 0 0 d = 5 r 1 ) r 2 ] Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap Ground states Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap How to quantify entanglement Entanglement of Motzkin States is due to the mutual information between halves k l a w n i k z t o M A B ( ( 0 ( ( 0 ) ) ) ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap How to quantify entanglement Entanglement of Motzkin States is due to the mutual information between halves k l a w n i k z t o M m A B ( ( 0 ( ( 0 ) ) ) ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap More than one type of ’parenthesis’ e.g., s = 2 Suppose there are two types ( and { to match { ( 0 { ( 0 } ) } ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap More than one type of ’parenthesis’ e.g., s = 2 Entanglement of Colored Motzkin States is due to the mutual information between halves k l a w n i k z t o M { ( 0 { ( 0 } ) } ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap Subtlety for s > 1 Suppose there are two types ( and { to match O.K. ( 0 { 0 } ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap Matching that does NOT work Suppose there are two types ( and { to match ( 0 { 0 ) } Not O.K. ! O.K. ( 0 { 0 } ) Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap The ground state s = 1 1 | p th Motzkin walk � √ M 2 n ∑ | M 2 n , s � = p e.g., 2 n = { 2 , 4 } 1 | M 2 � = 2 { | 00 � + | ℓ r �} 1 | M 4 � = 9 { | 0000 � + | 00 ℓ r � + | 0 ℓ 0 r � + | ℓ 00 r � + | 0 ℓ 0 r � + | ℓ 0 r 0 � + | ℓ r 00 � + | ℓ r ℓ r � + | ℓℓ rr �} Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
Quantifying entanglement, area laws and simulability The Local Hamiltonian and Gap The ground state s ≥ 1 1 | p th s-colored Motzkin walk � √ M 2 n ∑ M 2 n , s � = p e.g., 2 n = 2 � � s | ℓ k r k � ∑ M 2 , s � ∼ | 00 � + k = 1 Ramis Movassagh Movassagh and Shor: arXiv:1408.1657 [quant-ph]
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