entanglement area law (from heat capacity)
DESCRIPTION
Fernando G.S.L. Brand ão University College London Based on joint work arXiv:1410.XXXX with Marcus Cramer University of Ulm Isfahan, September 2014. Entanglement Area Law (from Heat Capacity). Plan. What is an area law? Relevance Previous Work Area Law from Heat Capacity. Area Law. - PowerPoint PPT PresentationTRANSCRIPT
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Entanglement Area Law (from Heat Capacity)
Fernando G.S.L. BrandãoUniversity College London
Based on joint work arXiv:1410.XXXX with
Marcus CramerUniversity of Ulm
Isfahan, September 2014
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Plan
• What is an area law?
• Relevance
• Previous Work
• Area Law from Heat Capacity
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Area Law
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Area Law
Quantum states on a lattice
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Area Law
Quantum states on a lattice
R
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Area Law
Quantum states on a lattice
Def: Area Law holds forif for all R,
R
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When does area law hold?
1st guess: it holds for every low-energy state of local models
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When does area law hold?
1st guess: it holds for every low-energy state of local models
(Irani ‘07, Gotesman&Hastings ‘07) There are 1D models with volume scaling of entanglement in groundstate
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When does area law hold?
1st guess: it holds for every low-energy state of local models
(Irani ‘07, Gotesman&Hastings ‘07) There are 1D models with volume scaling of entanglement in groundstate
Must put more restrictions on Hamiltonian/State!
Correlation length
spectral gap
specific heat
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Relevance
1D: Area LawSα, α < 1
Good ClassicalDescription
(MPS)
Renyi Entropies:
Matrix-Product-State:
(FNW ’91Vid ’04)
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Relevance
1D: Area LawSα, α < 1
Good ClassicalDescription
(MPS)
Renyi Entropies:
Matrix-Product-State:
> 1D: ???? (appears to be connected with good tensor network description; e.g. PEPS, MERA)
(FNW ’91Vid ’04)
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Previous Work(Bekenstein ‘73, Bombelli et al ‘86, ….) Black hole entropy
(Vidal et al ‘03, Plenio et al ’05, …) Integrable quasi-free bosonic systems and spin systems
see Rev. Mod. Phys. (Eisert, Cramer, Plenio ‘10)
…
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Previous Work(Bekenstein ‘73, Bombelli et al ‘86, ….) Black hole entropy
(Vidal et al ‘03, Plenio et al ’05, …) Integrable quasi-free bosonic systems and spin systems
see Rev. Mod. Phys. (Eisert, Cramer, Plenio ‘10)
2nd guess: Area Law holds for
1. Groundstates of gapped Hamiltonians
2. Any state with finite correlation length
…
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Gapped ModelsDef: (gap)
(gapped model) {Hn} gapped if
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Gapped ModelsDef: (gap)
(gapped model) {Hn} gapped if
gap
finitecorrelation length
(Has ‘04)
AB
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Gapped ModelsDef: (gap)
(gapped model) {Hn} gapped if
gap
finitecorrelation length
Exponential smallheat capacity
(Has ‘04)
(expectation no proof)
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Area Law?
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04) ξ < O(1/Δ)
Intuition: Finite correlation length should imply area law
X ZY
l = O(ξ)
(Uhlmann)
1D
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Area Law?
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04) ξ < O(1/Δ)
Intuition: Finite correlation length should imply area law
Obstruction: Data HidingX ZY
l = O(ξ)
1D
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Area Law?
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04)
???1D
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Area Law in 1D: A Success Story
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04)
(Hastings ’07) S < eO(1/Δ)
(Arad et al ’13) S < O(1/Δ)
ξ < O(1/Δ)
???
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Area Law in 1D: A Success Story
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04)
(Hastings ’07) S < eO(1/Δ)
(Arad et al ’13) S < O(1/Δ)
(B, Hor ‘13)S < eO(ξ) ξ < O(1/Δ)
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Area Law in 1D: A Success Story
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04)
(Hastings ’07) S < eO(1/Δ)
(Arad et al ’13) S < O(1/Δ)
(B, Hor ‘13)S < eO(ξ) ξ < O(1/Δ)
(Has ’07) Analytical (Lieb-Robinson bound, filtering function, Fourier analysis)(Arad et al ’13) Combinatorial (Chebyshev polynomial)(B., Hor ‘13) Information-theoretical (entanglement distillation, single-shot info theory)
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Area Law in 1D: A Success Story
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04)
(Hastings ’07) S < eO(1/Δ)
(Arad et al ’13) S < O(1/Δ)
(B, Hor ‘13)S < eO(ξ) ξ < O(1/Δ)
Efficient algorithm (Landau et al ‘14)
(Has ’07) Analytical (Lieb-Robinson bound, filtering function, Fourier analysis)(Arad et al ’13) Combinatorial (Chebyshev polynomial)(B., Hor ‘13) Information-theoretical (entanglement distillation, single-shot info theory)
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Area Law in 1D: A Success Story
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04)
(Hastings ’07) S < eO(1/Δ)
(Arad et al ’13) S < O(1/Δ)
(B, Hor ‘13)S < eO(ξ) ξ < O(1/Δ)
Efficient algorithm (Landau et al ‘14)
2nd guess: Area Law holds for
1. Groundstates of gapped Hamiltonians2. Any state with finite correlation length
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Area Law in 1D: A Success Story
gapfinite
correlation length
area law MPS(Has ‘04) (FNW ’91
Vid ’04)
(Hastings ’07) S < eO(1/Δ)
(Arad et al ’13) S < O(1/Δ)
(B, Hor ‘13)S < eO(ξ) ξ < O(1/Δ)
Efficient algorithm (Landau et al ‘14)
2nd guess: Area Law holds for
1. Groundstates of gapped Hamiltonians 1D, YES! >1D, OPEN 2. Any state with finite correlation length 1D, YES! >1D, OPEN
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Area Law from Specific HeatStatistical Mechanics 1.01
Gibbs state: ,
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Area Law from Specific HeatStatistical Mechanics 1.01
Gibbs state: ,
energy density:
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Area Law from Specific HeatStatistical Mechanics 1.01
Gibbs state: ,
energy density:
entropy density:
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Area Law from Specific HeatStatistical Mechanics 1.01
Gibbs state: ,
energy density:
entropy density:
Specific heat capacity:
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Area Law from Specific Heat
Specific heat at T close to zero:
Gapped systems: (superconductor, Haldane phase, FQHE, …)
Gapless systems:(conductor, …)
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Area Law from Specific Heat
Thm Let H be a local Hamiltonian on a d-dimensional lattice Λ := [n]d. Let (R1, …, RN), with N = nd/ld, be a partition of Λ into cubic sub-lattices of size l (and volume ld).
1. Suppose c(T) ≤ T-ν e-Δ/T for every T ≤ Tc. Then for every ψ with
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Area Law from Specific Heat
Thm Let H be a local Hamiltonian on a d-dimensional lattice Λ := [n]d. Let (R1, …, RN), with N = nd/ld, be a partition of Λ into cubic sub-lattices of size l (and volume ld).
2. Suppose c(T) ≤ Tν for every T ≤ Tc. Then for every ψ with
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Why?Free energy:
Variational Principle:
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Why?Free energy:
Variational Principle:
Let
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Why?Free energy:
Variational Principle:
Let
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Why?Free energy:
Variational Principle:
Let
By the variational principle, for T s.t. u(T) ≤ c/l :
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Why?Free energy:
Variational Principle:
Let
By the variational principle, for T s.t. u(T) ≤ c/l :
Result follows from:
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Summary and Open Questions
Summary:
Assuming the specific heat is “natural”, area law holds for every low-energy state of gapped systems and “subvolume law” for every low-energy state of general systems
Open questions:
- Can we prove a strict area law from the assumption on c(T) ≤ T-ν e-Δ/T ?
- Can we improve the subvolume law assuming c(T) ≤ Tν ?
- Are there natural systems violating one of the two conditions?
- Prove area law in >1D under assumption of (i) gap (ii) finite correlation length
- What else does an area law imply?