Renyi Entropies from Random Quenches in Atomic Hubbard and Spin - PowerPoint PPT Presentation

24.10.17 Cold Quantum Coffee HD 24.8.2017 Renyi Entropies from Random Quenches in Atomic Hubbard and Spin Models Andreas Elben with B. Vermersch, M. Dalmonte, I. Cirac and P. Zoller arXiv: 1709.05060 University of Innsbruck/IQOQI UQUAM ERC Synergy Grant UNIVERSITY OF INNSBRUCK

Outline 1.Why Renyi entropies and Entanglement? Which setups? A B 2. Measurement of Renyi entropies in atomic Hubbard and Spin models 3. Examples - Area Law and Many-Body Localization

Renyi entropies as an entanglement measure Two subsystems A and B are bipartite entangled iff A B | Ψ AB i 6 = | Ψ A i ⌦ | Ψ B i Sufficient condition for bipartite entanglement ρ A = Tr B | ψ AB i h ψ AB | Reduced density matrix | {z } ρ AB ⇥ ρ 2 ⇤ ⇥ ρ 2 ⇤ Purity of subsystem Tr < Tr Purity of full system A AB Entanglement entropies S A = − Tr A [ ρ A log ρ A ] von-Neumann 1 S ( n ) 1 − n log Tr A [ ρ n = A ] ≤ S A Renyi A

Why Entanglement? Entanglement as a Entanglement as a characterising property resource - - Quantum many body systems Quantum computing Scaling with (sub-)sytem size - Spreading of quantum correlations, Trapped ion quantum computer Topological phases, … Spin triplet c. d. e. 2 0 . 6 DMRG 4 S (2) / ∂ A 0 . 4 Area law Spin 6 8 0 . 2 10 12 0 . 0 Lab ZZ Lab XY Lab XY Lab YY Lab YY 14 2 4 6 8 2 ∂ A 4 Dynamics - Thermalisation vs. Spin 6 8 Many-Body Localization 10 12 MPS ZZ MPS XY MPS YY 14 0 . 6 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 U/J = 1 , ∆ /J = 10 Spin Spin Spin -0.5 0.5 U/J = 0 , ∆ /J = 10 0 . 5 S (2) Lanyon et al., arXiv:1612.08000 0 . 4 0 . 3 10 0 10 1 10 2 Jt

Scaling of entanglement B Volume law - Extensive scaling S ( ρ A ) ∼ | A | • ‘generic’ (random) quantum states • thermal states e.g. Berges et al., arXiv:1707.05338 Area Law S ( ρ A ) ∼ | ∂ A | ∂ A A • ground states of typical (local, gapped) Hamiltonians ρ A = Tr B [ ρ ] Holographic principle and Topological entanglement Complexity of Simulations Black holes entropy - MPS, PEPS Bekenstein- Hawking: (Logarithmic) corrections Success of DMRG in 1D to area laws systems S BH = horizon area 4 Srednicki: Massless scalar fields Eisert et al., Rev. Mod. Phys. 82, 277 (2010)

Area Law in a Heisenberg model Isotropic Heisenberg model (here: 2D) X l + σ y i σ y σ x i σ x l + σ z i σ z H h = J l h il i A Ground state ρ GS ∂ A S (2) ( A ) = − log Tr A ρ 2 ⇥ ⇤ A 8x8 sites with ρ A = Tr S \ A [ ρ GS ] 0 . 6 How to verify/measure this in an DMRG S (2) / ∂ A experiment? 0 . 4 • Physical realization? 0 . 2 • Measurement of entanglement? Area law 0 . 0 Quantum simulation 2 4 6 8 ∂ A

Examples of atomic quantum simulators Ultracold atoms in optical lattices Rydberg Atoms in optical tweezers (MPQ, HD, CUA, JQI, …) (MPQ, CUA, IOGS,…) Endres et al., Science (2016) Barredo et al., Science (2016) Kaufman et al., Science (2016) Choi et al., Science (2016) bosonic/fermionic Hubbard models Spin (Ising) models,… Heisenberg model,… Trapped Ions (IBK, JQI, Oxford,…) How to measure Renyi entropies demonstrating entanglement? R. Blatt, Innsbruck Spin (Ising) models, …

Measurement of Renyi entropies Quantum State Tomography b. 0.5 Gross et al. PRL 105, 0.25 150401 (2010) 0 B. P. Lanyon et al., Nat.Phys., ↑↑ ↑↓ (2017) 1 ↓↑ ↓↓ ↓↓ ↓↑ 0.5 If available ↑↓ ↑↑ Quench dynamics, Interference of copies Adiabatic preparation, Initial state … Islam et al., Nature 528, 77–83 (2015) A B Daley et al . PRL, 109(2), 20505 (2012) ρ → ρ A ρ I A odd or even mixed Pichler et al. PRX, 6(4), 41033 A+B even pure (2016) Measurement of A and B entangled Renyi entropy of a (sub-)system Random measurements on single ∼ Tr [ ρ n A ] copies in a QC Realization in a atomic quantum many-body van Enk, Beenaker, PRL 108, 110503 (2012) systems existing today in the lab?! 
 S. Boixo, et al., arXiv:1608.00263 Hubbard/Spin models

Outline 1.Why Renyi entropies and Entanglement? Which systems? A B 2. Measurement of Renyi entropies in atomic Hubbard and Spin models • Mini-Review: Entanglement from random measurements van Enk, Beenaker (PRL 2012) • Physical Realization 3. Examples - Area Law and MBL

Random measurements & Quantum information Protocol for a chain of qubits: A ` van Enk, Beenaker (PRL 2012) ρ A Random measurement random unitary U A by random gates h i U A ρ A U † U A ρ A U † P ( s A ) = Tr A | s A i h s A | ρ A ` A Measurement Average over the Circular Unitary Ensemble (CUE) of qubit states ( 0 , 1 , 0 ) = s A ⇥ ρ 2 ⇤ 1 1 + Tr A h P ( s A ) i = h P ( s A ) 2 i = N H A N H A ( N H A + 1) Hilbertspace dimension of A

Random measurements & Quantum information Random unitaries from the Circular Unitary Ensemble (CUE) Unitaries distributed according the Haar measure on the unitary group : Hilbert space ✓ ◆ N H A 1 U A ∈ CUE( N H A ) < ( U ij ) , = ( U ij ) ⇠ N 0 , dimension of subsystem N H A up to unitary constraints Projector Application to the protocol: describing measurement Measurement h i U A ρ A U † ρ f A = U ρ A U † P ( s A ) = Tr A P s A ρ A with outcome s A Virtual copies i = Tr [ ρ A ] 2 + Tr ⇥ ⇤ ρ 2 h i . . . U A ρ A U † A ⌦ U A ρ A U † A h P( s A ) 2 i = h Tr 1 ⊗ 2 A . . . N H A ( N H A + 1) in i = δ kl δ mn + δ kn δ ml CUE (2-design) : h U ik U ∗ il U im U ∗ N H A ( N H A + 1) ~ Gaussian

Random measurements & Quantum information Protocol for a chain of qubits: A ` van Enk, Beenaker (PRL 2012) ρ A Random measurement random unitary U A by random gates h i U A ρ A U † P ( s A ) = Tr A | s A i h s A | ` Measurement Average over the Circular Unitary Ensemble (CUE) of qubit states ( 0 , 1 , 0 ) = s A ⇥ ρ 2 ⇤ 1 1 + Tr A h P ( s A ) i = h P ( s A ) 2 i = N H A N H A ( N H A + 1) Hilbertspace dimension of A Realization in a Hubbard or Spin model: How many measurements How to generate random per unitary and how many unitaries? unitaries?

Outline 1.Why entanglement? How to quantify? A B 2. Measurement of Renyi entropies in atomic Hubbard and Spin models • Mini-Review: Entanglement from random measurements van Enk, Beenaker (PRL 2012) • Physical Realization 3. Examples - Area Law and MBL

Measurement protocol for Hubbard and Spin models Quench dynamics, Adiabatic preparation,… Time ρ → ρ A ρ I Measurement of Renyi entropy of a (sub-)system ∼ Tr [ ρ n A ]

Measurement protocol for Hubbard and Spin models Random unitary as time evolution operator under random quenches U A = e − iH η T · · · e − iH 1 T Quench dynamics, 1 η Adiabatic preparation,… U A ρ A U † Time ρ → ρ A ρ I A Disorder ↑ pattern ↓ potential offsets See also: M Ohliger, V Nesme, J Eisert - NJP 2013

Generation of random unitaries Idea : Random unitary as time evolution operator 
 resulting from a series of random quenches X Heisenberg model (e.g. strong interaction limit of FH) H h = J σ i · σ l h il i2 A Disorder patterns ∆ j X i σ z H j = H h + i i ∈ A ∆ j from gaussian distribution i with standard deviation ∆ potential offsets U A = e − iH η T · · · e − iH 1 T Vary disorder in discrete steps in time Random unitary?

Generation of random unitaries U A = e − iH η T · · · e − iH 1 T Question: Is a random unitary? Apply the protocol to a known input state and compare estimated to true purity to test the ensemble Heisenberg model (here: 1D, 8 sites) X X ∆ j i σ z H j = J σ i · σ l + i AF 10 1 10 i 2 A h il i2 A PS ( p 2 ) e ∆ j from gaussian distribution 
 Rand i with standard deviation ∆ = AF + PS 10 0 J = ∆ = 1 /T 1 . 0 exponential 0 . 5 AF: | "#"#"#"#i convergence 32 0 16 PS: | """"####i η Rand: random pure state Random unitaries using ✓ generic interactions ✓ engineered disorder

Scaling with system size Heisenberg model with L sites X X ∆ j i σ z H j = J σ i · σ l + i U A = e − iH η T · · · e − iH 1 T i 2 A h il i2 A ∆ j from gaussian distribution 
 i with standard deviation ∆ = 1D 2D N U = 500 N U = 500 10 1 10 1 | ( p 2 ) e − p 2 | | ( p 2 ) e − p 2 | 2 8 2 × 2 3 × 3 10 0 10 0 4 10 3 × 2 5 × 2 L = L x × L y 4 × 2 6 10 − 1 10 − 1 10 − 2 10 − 2 0 2 4 6 8 0 2 4 6 8 10 η /L η /L Number of necessary statistical error threshold due to finite number random quenches (500) random unitaries η ∼ L Efficient generation of random unitaries for purity measurements Random quantum h P ( s A ) 2 i ⇠ Tr ρ 2 ⇥ ⇤ A circuits

Ising and Hubbard models Hubbard models Quantum Ising models (Bosons/Fermions) (Rydberg atoms / Ions) C 6 r 6 J ∆ 2 nP 3 / 2 U ∆ 4 ∆ 3 ∆ 1 Ω 87 Rb Ω Ω Ω 5 S 1 / 2 2 ∆ Ω Ω Ω Ω C 6 + U ⇣ ⌘ X X ∆ j X X a † X X ∆ j σ x i σ z | r i − r j | 6 σ z i σ z H j = Ω H j = − J i +1 a i + h.c. n i ( n i − 1) + i n i i + i + j 2 i ∈ A i ∈ A i ∈ A i i i<j Ω = C 6 /a 6 = ∆ = 1 /T U = J = ∆ = 1 /T L = 8 , N = 4 L = 8 10 0 | ( p 2 ) e − p 2 | | ( p 2 ) e − p 2 | Fock States Ground state 10 1 Ground State Random state Random State Fock states 10 − 1 10 − 1 Random unitaries created with existing tools 10 − 3 ✓ generic interactions 1 2 3 0 2 η /L ✓ engineered disorder η /L