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

UQUAM

ERC Synergy Grant

UNIVERSITY OF INNSBRUCK

Andreas Elben with B. Vermersch, M. Dalmonte, I. Cirac and P. Zoller University of Innsbruck/IQOQI

24.8.2017 24.10.17 Cold Quantum Coffee HD

Renyi Entropies from Random Quenches in Atomic Hubbard and Spin Models

arXiv: 1709.05060

Outline

1.Why Renyi entropies and Entanglement? Which setups?

• 2. Measurement of Renyi entropies in atomic Hubbard and

Spin models

• 3. Examples - Area Law and Many-Body Localization

A B

Renyi entropies as an entanglement measure

Two subsystems A and B are bipartite entangled iff |ΨABi 6= |ΨAi ⌦ |ΨBi

A

B

Entanglement entropies

SA = −TrA [ρA log ρA]

von-Neumann Renyi

S(n)

A

= 1 1 − n log TrA [ρn

A]

≤ SA

Tr ⇥ ρ2

A

⇤ < Tr ⇥ ρ2

AB

⇤ Purity of subsystem Purity of full system Reduced density matrix Sufficient condition for bipartite entanglement ρA = TrB|ψABi hψAB| | {z }

ρAB

Why Entanglement?

Entanglement as a resource

• Quantum computing

Spin triplet

c. d. e.

2 4 6 8 10 12 14 Spin 2 4 6 8 10 12 14 Spin 2 4 6 8 10 12 14 Spin 2 4 6 8 10 12 14 Spin 2 4 6 8 10 12 14 Spin

• 0.5

0.5 Lab ZZ MPS ZZ MPS YY Lab YY MPS XY Lab XY Lab YY Lab XY

Lanyon et al., arXiv:1612.08000

Trapped ion quantum computer

Scaling with (sub-)sytem size - Spreading of quantum correlations, Topological phases, …

Entanglement as a characterising property

• Quantum many body systems

2 4 6 8

∂A

0.0 0.2 0.4 0.6

S(2)/∂A

DMRG

Dynamics - Thermalisation vs. Many-Body Localization

100 101 102

Jt

0.3 0.4 0.5 0.6

S(2)

U/J = 1, ∆/J = 10 U/J = 0, ∆/J = 10

Area law

Scaling of entanglement

A

∂A

Volume law - Extensive scaling

• ‘generic’ (random) quantum states
• thermal states

S (ρA) ∼ |A|

Area Law

• ground states of typical (local, gapped)

Hamiltonians

S (ρA) ∼ |∂A|

ρA = TrB [ρ]

Holographic principle and Black holes Topological entanglement entropy Complexity of Simulations

• MPS, PEPS

SBH = horizon area 4

Bekenstein- Hawking: Srednicki: Massless scalar fields Success of DMRG in 1D systems (Logarithmic) corrections to area laws Eisert et al., Rev. Mod. Phys. 82, 277 (2010)

e.g. Berges et al., arXiv:1707.05338

B

Area Law in a Heisenberg model

Hh = J X

hili

σx

i σx l + σy i σy l + σz i σz l

Isotropic Heisenberg model (here: 2D) Ground state ρGS S(2)(A) = − log TrA ⇥ ρ2

A

⇤ with ρA = TrS\A [ρGS]

2 4 6 8

∂A

0.0 0.2 0.4 0.6

S(2)/∂A

DMRG

Area law 8x8 sites

How to verify/measure this in an experiment?

• Physical realization?
• Measurement of entanglement?

Quantum simulation

A ∂A

Examples of atomic quantum simulators

How to measure Renyi entropies demonstrating entanglement? Rydberg Atoms in optical tweezers (MPQ, CUA, IOGS,…)

Endres et al., Science (2016) Barredo et al., Science (2016)

Trapped Ions (IBK, JQI, Oxford,…)

• R. Blatt, Innsbruck

Ultracold atoms in optical lattices (MPQ, HD, CUA, JQI, …)

Kaufman et al., Science (2016) Choi et al., Science (2016)

bosonic/fermionic Hubbard models Heisenberg model,… Spin (Ising) models, … Spin (Ising) models,…

Measurement of Renyi entropies ρI

Initial state Interference of copies

Daley et al . PRL, 109(2), 20505 (2012) Pichler et al. PRX, 6(4), 41033 (2016) Islam et al., Nature 528, 77–83 (2015)

A B A+B even pure A odd or even mixed A and B entangled

Measurement of Renyi entropy

• f a (sub-)system

Random measurements on single copies in a QC

van Enk, Beenaker, PRL 108, 110503 (2012)

• S. Boixo, et al., arXiv:1608.00263

Realization in a atomic quantum many-body systems existing today in the lab?!
 Hubbard/Spin models Quantum State Tomography

0.5 1

0.5 0.25

↑↑ ↑↓ ↓↑ ↓↓ ↑↑ ↑↓ ↓↑ ↓↓

b.

• B. P. Lanyon et al., Nat.Phys.,

(2017) Gross et al. PRL 105, 150401 (2010)

If available

ρ → ρA

∼ Tr [ρn

A]

Quench dynamics, Adiabatic preparation, …

Outline

1.Why Renyi entropies and Entanglement? Which systems?

• 2. Measurement of Renyi entropies in atomic Hubbard and

Spin models

• Mini-Review: Entanglement from random measurements
• Physical Realization
• 3. Examples - Area Law and MBL

A B

van Enk, Beenaker (PRL 2012)

Random measurements & Quantum information

Protocol for a chain of qubits: A

`

Measurement

• f qubit states

( 0 , 1 , 0 ) = sA

`

UA

random unitary by random gates

Random measurement ρA

UAρAU †

A

P(sA) = Tr h UAρAU †

A |sAi hsA|

i ρA

van Enk, Beenaker (PRL 2012)

Average over the Circular Unitary Ensemble (CUE)

hP(sA)i = 1 NHA hP(sA)2i = 1 + Tr ⇥ ρ2

A

⇤ NHA(NHA + 1)

Hilbertspace dimension of A

Random measurements & Quantum information

<(Uij), =(Uij) ⇠ N ✓ 0, 1 NHA ◆

up to unitary constraints UA ∈ CUE(NHA) Random unitaries from the Circular Unitary Ensemble (CUE)

: Hilbert space dimension of subsystem

Unitaries distributed according the Haar measure on the unitary group ρA ρf

A = UρAU †

P (sA) = Tr h UAρAU †

APsA

i

Measurement with outcome Projector describing measurement sA

Application to the protocol:

CUE (2-design) : hUikU ∗

ilUimU ∗ ini = δklδmn + δknδml

NHA(NHA + 1)

~ Gaussian

hP(sA)2i = hTr1⊗2 h . . . UAρAU †

A⌦UAρAU † A . . .

i i = Tr [ρA]2 + Tr ⇥ ρ2

A

⇤ NHA(NHA + 1)

Virtual copies

NHA

Random measurements & Quantum information

Protocol for a chain of qubits: A

`

Measurement

• f qubit states

( 0 , 1 , 0 ) = sA

`

UA

random unitary by random gates

Random measurement ρA

P(sA) = Tr h UAρAU †

A |sAi hsA|

i

van Enk, Beenaker (PRL 2012)

Average over the Circular Unitary Ensemble (CUE)

hP(sA)i = 1 NHA hP(sA)2i = 1 + Tr ⇥ ρ2

A

⇤ NHA(NHA + 1)

Hilbertspace dimension of A

Realization in a Hubbard or Spin model:

How to generate random unitaries? How many measurements per unitary and how many unitaries?

Outline

1.Why entanglement? How to quantify?

• 2. Measurement of Renyi entropies in atomic Hubbard and

Spin models

• Mini-Review: Entanglement from random measurements
• Physical Realization
• 3. Examples - Area Law and MBL

A

B

van Enk, Beenaker (PRL 2012)

Measurement protocol for Hubbard and Spin models ρI

Time

ρ → ρA

Quench dynamics, Adiabatic preparation,…

Measurement of Renyi entropy

• f a (sub-)system

∼ Tr [ρn

A]

Measurement protocol for Hubbard and Spin models ρI

Random unitary as time evolution operator under random quenches

Time

ρ → ρA UAρAU †

A

UA = e−iHηT · · · e−iH1T

1 η

See also: M Ohliger, V Nesme, J Eisert - NJP 2013

Disorder pattern potential

• ffsets

↑

↓

Quench dynamics, Adiabatic preparation,…

Generation of random unitaries

Idea: Random unitary as time evolution operator 
 resulting from a series of random quenches

UA = e−iHηT · · · e−iH1T

Heisenberg model (e.g. strong interaction limit of FH) Vary disorder in discrete steps in time Random unitary? from gaussian distribution with standard deviation ∆ ∆j

i

Hj = Hh + X

i∈A

∆j

i σz i

Hh = J X

hili2A

σi · σl

Disorder patterns potential

• ffsets

Generation of random unitaries

Apply the protocol to a known input state and compare estimated to true purity to test the ensemble Question: Is a random unitary?

UA = e−iHηT · · · e−iH1T

Random unitaries using ✓ generic interactions ✓ engineered disorder Hj = J X

hili2A

σi · σl + X

i2A

∆j

i σz i

from gaussian distribution 
 with standard deviation ∆j

i

∆ =

Heisenberg model (here: 1D, 8 sites)

exponential convergence

AF: |"#"#"#"#i PS: |""""####i J = ∆ = 1/T

100 101

AF PS Rand AF + PS

η

0.5 1.0 10

(p2)e

Rand: random pure state

16 32

Scaling with system size

Heisenberg model with L sites 1D Number of necessary random quenches

η ∼ L

2 4 6 8 10−2 10−1 100 101

|(p2)e − p2|

2 4 6 8 10

η/L

2D

2 4 6 8 10 10−2 10−1 100 101

|(p2)e − p2|

2 × 2 3 × 2 4 × 2 3 × 3 5 × 2

η/L

L = Lx × Ly

NU = 500

UA = e−iHηT · · · e−iH1T

Hj = J X

hili2A

σi · σl + X

i2A

∆j

i σz i

from gaussian distribution 
 with standard deviation ∆j

i

∆ =

statistical error threshold due to finite number (500) random unitaries NU = 500

Efficient generation of random unitaries for purity measurements

hP(sA)2i ⇠ Tr ⇥ ρ2

A

⇤

Random quantum circuits

Ising and Hubbard models

J

U

2∆

Hubbard models (Bosons/Fermions)

Hj = −J X

i∈A

⇣ a†

i+1ai + h.c.

⌘ + U 2 X

i∈A

ni(ni − 1) + X

i∈A

∆j

ini

U = J = ∆ = 1/T

2

η/L

10−3 10−1 101

|(p2)e − p2|

Fock States Ground State Random State

L = 8, N = 4

87Rb

nP3/2

Ω

Ω

Ω Ω Ω

∆1 ∆2 ∆3 ∆4

Ω Ω Ω

C6 r6 5S1/2

Quantum Ising models (Rydberg atoms / Ions)

Hj = Ω X

i

σx

i +

X

i

∆j

iσz i +

X

i<j

C6 |ri − rj|6 σz

i σz j

Ω = C6/a6 = ∆ = 1/T

1 2 3

η/L

10−1 100

|(p2)e − p2|

Ground state Random state Fock states

L = 8

Random unitaries created with existing tools ✓ generic interactions ✓ engineered disorder

Measurement protocol for Hubbard and Spin models ρI

Random unitary as time evolution operator under random quenches

Time

ρ → ρA UAρAU †

A

UA = e−iHηT · · · e−iH1T

1 η

Disorder pattern potential

• ffsets

↑

↓

Quench dynamics, Adiabatic preparation,…

Measurement protocol for Hubbard and Spin models

ρI

State preparation

Random unitary

Time

ρ → ρA

Disorder patterns Potential
 barriers

↑

↓

UAρAU †

A

Quantum Gas Microscope

(n↑, n↓)

Measurement

Quench dynamics, Adiabatic preparation,…

Repeat scheme correlations For many random unitaries hP(n↑, n↓)2i ⇠ Tr ⇥ ρ2

A

⇤ For the same random unitary P(n↑, n↓) probabilities for all measurement outcomes

(n↑, n↓)

Scaling of statistical errors

Error for estimated purity (averaged over all outcomes) : number of measurements per unitary : number of unitaries : Hilbert space dimension of A

NM

NU

NHA

101 102 103

NU

10−2 10−1 100

|(p2)e − p2|

√NA NA ∞

NM = ∼ 1 √NU

100 101 102

√

10−2 10−1 100

|(p2)e − p2|

64 128 256

NM/ p NHA

NHA

NU = 1000

∼ p NHA NM |(p2)e − p2| ∼ 1 p NUNHA ✓ 1 + NHA NM ◆ Analytics: finiteNU finiteNM

NHA = 256

Scaling of statistical errors

Error for estimated purity (averaged over all outcomes) : number of measurements per unitary : number of unitaries : Hilbert space dimension of A

NM

NU

NHA

101 102 103

NU

10−2 10−1 100

|(p2)e − p2|

√NA NA ∞

NM = ∼ 1 √NU

100 101 102

√

10−2 10−1 100

|(p2)e − p2|

64 128 256

NM/ p NHA

NHA

NU = 1000

∼ p NHA NM

NHA = 256

Number of measurements to determineasf
 up to error sdsdfgss

NM ∼ p NHA

p2

∼ 1/ p NU

Outline

1.Why Renyi entropies and Entanglement? Which systems?

• 2. Measurement of Renyi entropies in atomic Hubbard and

Spin models

• Mini-Review: Entanglement from random measurements
• Physical Realization
• 3. Examples - Area Law and MBL

van Enk, Beenaker (PRL 2012)

Area Law in a Heisenberg model

Hh = J X

hili

σx

i σx l + σy i σy l + σz i σz l

Isotropic Heisenberg model (here: 2D) Ground state ρGS S(2)(A) = − log TrA ⇥ ρ2

A

⇤ with ρA = TrS\A [ρGS]

2 4 6 8

∂A

0.0 0.2 0.4 0.6

S(2)/∂A

DMRG

2 4 6 8

∂A

0.0 0.2 0.4 0.6

S(2)/∂A

DMRG 1 2 4 8

η

NM = 100 NU = 100

8x8 sites

A ∂A

Entanglement dynamics: Many-Body Localization

Many Body Localization A quantum phase characterized by the interplay of disorder (localization) and interactions Only known generic exception of thermalisation (ETH)

Rahul Nandkishore, David A. HuseAnnual Review of Condensed Matter Physics, Vol. 6: 15-38 (2015) D.M. Basko, I.L. Aleiner, B.L. Altshuler Annals of Physics 321, 1126 (2006)

Von Neumann Entropy Time Anderson Localization MBL

Hallmark of MBL: slow (logarithmic, but nonzero) growth of entanglement No direct observation

• f entanglement

growth so far!

Bardarson, et al., PRL 2012

Entropy growth in the many-body localised phase

J

U

2∆

Bose Hubbard model (1D)

HBH = −J X

i

⇣ a†

iai + h.c.

⌘ + U 2 X

i

ni(ni − 1) + ∆i ∈ [−∆, ∆] 1) + X

i

∆ini

Disorder:

10−1 100 101 102

Jt

10−2 10−1 100 101

S(2)

U = J, ∆ = 0

Quench dynamics of entanglement

power laws ‘Thermalization’

L = 10, N = 5

Initial state: |1010101010i

Entropy growth in the many-body localised phase

J

U

2∆

Bose Hubbard model (1D)

HBH = −J X

i

⇣ a†

iai + h.c.

⌘ + U 2 X

i

ni(ni − 1) +

10−1 100 101 102

Jt

10−2 10−1 100 101

S(2)

U = J = ∆ U = J, ∆ = 0

∆i ∈ [−∆, ∆] 1) + X

i

∆ini

Disorder: Quench dynamics of entanglement

‘Thermalization’

L = 10, N = 5

Initial state: |1010101010i

‘Localization’

Entropy growth in the many-body localised phase

J

U

2∆

Bose Hubbard model (1D)

HBH = −J X

i

⇣ a†

iai + h.c.

⌘ + U 2 X

i

ni(ni − 1) +

10−1 100 101 102

Jt

10−1 101

S(2)

U/J = 1, ∆/J = 10 U/J = 0, ∆/J = 10 U/J = 1, ∆/J = 0

∆i ∈ [−∆, ∆] 1) + X

i

∆ini

Disorder: Quench dynamics of entanglement

L = 10, N = 5

Initial state: |1010101010i

‘Thermalization’ ‘Localization’

Entropy growth in the many-body localised phase

J

U

2∆

Bose Hubbard model (1D)

HBH = −J X

i

⇣ a†

iai + h.c.

⌘ + U 2 X

i

ni(ni − 1) +

10−1 100 101 102

Jt

10−1 101

S(2)

U/J = 1, ∆/J = 10 U/J = 0, ∆/J = 10 U/J = 1, ∆/J = 0

∆i ∈ [−∆, ∆] 1) + X

i

∆ini

Disorder: Quench dynamics of entanglement

100 101 102

Jt

0.3 0.4 0.5 0.6

S(2)

U/J = 1, ∆/J = 10 U/J = 0, ∆/J = 10

MBL Anderson Localization

L = 10, N = 5

Initial state: |1010101010i

‘Thermalization’ ‘Localization’

Entropy growth - Thermalization vs MBL

J

U

2∆

Bose Hubbard model (1D)

HBH = −J X

i

⇣ a†

iai + h.c.

⌘ + U 2 X

i

ni(ni − 1) +

100 101 102

Jt

0.3 0.4 0.5 0.6

S(2)

U/J = 1, ∆/J = 10 U/J = 0, ∆/J = 10

10−1 100 101 102

Jt

10−1 101

S(2)

U/J = 1, ∆/J = 10 U/J = 0, ∆/J = 10 U/J = 1, ∆/J = 0

NM = 100 NU = 100

∆i ∈ [−∆, ∆] 1) + X

i

∆ini

Disorder: Quench dynamics of entanglement

MBL Anderson localised

L = 10, N = 5

Initial state: |1010101010i

Entropy growth in the many-body localised phase

‘Thermalization’ ‘Localization’

Conclusion Thank you very much for your attention!

Measurement of Renyi entropies on single copies in atomic Hubbard and Spin models

• Renyi entropies can be inferred from random

measurements


• Uses only existing tools:

✓ Generic interactions ✓ Engineered disorder ✓ Quantum Gas microscope

• Application to systems of moderate size

• Variant: Protocol based on local unitaries

Rydberg atoms Hubbard models Ion Traps arXiv: 1709.05060