Multipartite entanglement certification in quantum many-body systems - - PowerPoint PPT Presentation

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Multipartite entanglement certification in quantum many-body systems - - PowerPoint PPT Presentation

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Multipartite entanglement certification in quantum many-body systems using quench dynamics Ricardo Costa de Almeida Institute for

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Multipartite entanglement certification in quantum many-body systems using quench dynamics

Ricardo Costa de Almeida

Institute for Theoretical Physics Heidelberg University Department of Physics University of Trento

Cold Quantum Coffee - 19/11/2019

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Contents

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Phase Estimation

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Phase Estimation

Goal: estimate a parameter θ =?

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Phase Estimation

Goal: estimate a parameter θ =? Tools: measurements of a quantum state ρ(θ) = e−iθOρ0e+iθO

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Phase Estimation

Goal: estimate a parameter θ =? Tools: measurements of a quantum state ρ(θ) = e−iθOρ0e+iθO How precise can this estimation be?

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Cram´ er-Rao Bound

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Cram´ er-Rao Bound

Conditional probability distribution: f (µ|θ)

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Cram´ er-Rao Bound

Conditional probability distribution: f (µ|θ) Calculating θ from outcomes of µ yields an estimator ˆ θ = ˆ θ(µ)

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Cram´ er-Rao Bound

Conditional probability distribution: f (µ|θ) Calculating θ from outcomes of µ yields an estimator ˆ θ = ˆ θ(µ) Fisher information: F =

  • µ

f (µ|θ) (∂θ ln f (µ|θ))

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Cram´ er-Rao Bound

Conditional probability distribution: f (µ|θ) Calculating θ from outcomes of µ yields an estimator ˆ θ = ˆ θ(µ) Fisher information: F =

  • µ

f (µ|θ) (∂θ ln f (µ|θ)) Bound on the precision of any estimator: Var (ˆ θ) ≥ F −1

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Cram´ er-Rao Bound

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Cram´ er-Rao Bound

Parameter-dependent quantum state: ρ(θ) = e−iθOρ0e+iθO

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Cram´ er-Rao Bound

Parameter-dependent quantum state: ρ(θ) = e−iθOρ0e+iθO Given some measurement setup: POVM{Eµ}

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Cram´ er-Rao Bound

Parameter-dependent quantum state: ρ(θ) = e−iθOρ0e+iθO Given some measurement setup: POVM{Eµ} ⇒ f (µ|θ) = Tr (ρ(θ)Eµ)

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Cram´ er-Rao Bound

Parameter-dependent quantum state: ρ(θ) = e−iθOρ0e+iθO Given some measurement setup: POVM{Eµ} ⇒ f (µ|θ) = Tr (ρ(θ)Eµ) ⇒ F({Eµ})

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Cram´ er-Rao Bound

Parameter-dependent quantum state: ρ(θ) = e−iθOρ0e+iθO Given some measurement setup: POVM{Eµ} ⇒ f (µ|θ) = Tr (ρ(θ)Eµ) ⇒ F({Eµ}) Quantum Fisher information: FQ = max

{Eµ} F ({Eµ})

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quantum Cram´ er-Rao Bound

Parameter-dependent quantum state: ρ(θ) = e−iθOρ0e+iθO Given some measurement setup: POVM{Eµ} ⇒ f (µ|θ) = Tr (ρ(θ)Eµ) ⇒ F({Eµ}) Quantum Fisher information: FQ = max

{Eµ} F ({Eµ})

Best precision achievable with ρ0: Var (ˆ θ) ≥ F −1

Q

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

How to Calculate the QFI?

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

How to Calculate the QFI?

Pure states ρ0 = |ψ ψ|:

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

How to Calculate the QFI?

Pure states ρ0 = |ψ ψ|: FQ = 4Var (O, ψ) = 4

  • ψ| O2 |ψ − ψ| O |ψ2
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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

How to Calculate the QFI?

Pure states ρ0 = |ψ ψ|: FQ = 4Var (O, ψ) = 4

  • ψ| O2 |ψ − ψ| O |ψ2

Mixed states ρ0 =

λ ρλ |λ λ|:

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

How to Calculate the QFI?

Pure states ρ0 = |ψ ψ|: FQ = 4Var (O, ψ) = 4

  • ψ| O2 |ψ − ψ| O |ψ2

Mixed states ρ0 =

λ ρλ |λ λ|:

FQ = 2

  • λ,λ′

ρλ − ρλ′ ρλ + ρλ′ (ρλ − ρλ′) | λ| O |λ′ |2 ,

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Multipartite Entanglement

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Multipartite Entanglement

System of N spins 1/2 |ψ ∈ H1 ⊗ · · · ⊗ HN

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Multipartite Entanglement

System of N spins 1/2 |ψ ∈ H1 ⊗ · · · ⊗ HN Product states: |ψ = |φ1 ⊗ · · · ⊗ |φN

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Multipartite Entanglement

System of N spins 1/2 |ψ ∈ H1 ⊗ · · · ⊗ HN Product states: |ψ = |φ1 ⊗ · · · ⊗ |φN k-producible states: |ψ = |ψi1 ⊗ · · · ⊗ |ψiP where each |ψip is a state of at most k spins

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Multipartite Entanglement

System of N spins 1/2 |ψ ∈ H1 ⊗ · · · ⊗ HN Product states: |ψ = |φ1 ⊗ · · · ⊗ |φN k-producible states: |ψ = |ψi1 ⊗ · · · ⊗ |ψiP where each |ψip is a state of at most k spins ◮ Entangled states = product states ◮ k-partite entangled states = k-producible states

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

QFI as an Entanglement Witness

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

QFI as an Entanglement Witness

For a k-producible state |ψ = |ψi1 ⊗ · · · ⊗ |ψiP and O =

j Oj:

Var (O, ψ) =

  • ip

Var (Oip, ψip)

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

QFI as an Entanglement Witness

For a k-producible state |ψ = |ψi1 ⊗ · · · ⊗ |ψiP and O =

j Oj:

Var (O, ψ) =

  • ip

Var (Oip, ψip) This leads to bounds for the FQ of k-producible states: FQ ≤ kN for O = 1 2

  • j

σz

j

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

QFI as an Entanglement Witness

For a k-producible state |ψ = |ψi1 ⊗ · · · ⊗ |ψiP and O =

j Oj:

Var (O, ψ) =

  • ip

Var (Oip, ψip) This leads to bounds for the FQ of k-producible states: FQ ≤ kN for O = 1 2

  • j

σz

j

k-partite entanglement certifcation: FQ > kN ⇒ k-partite entanglement

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

QFI in the thermal ensemble (BLACKBOARD)

FQ [ρ, O] = 4 π ∞ dω tanh ω 2T

  • χ”(ω, T)

FQ [ρ, O] = 4T ∞ dt χ(t, T) sinh (πtT) QFI from quench dynamics FQ [ρ, O] = 4πT 2 q ∞ dt ∆O(t)quench sinh (πtT) tanh (πtT)

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Quench Protocol

2 4 20 40 60

J a)

U/J=-4 U/J=-2 2 4

J t

5 10 15

J b)

2 4 6 8 20 40

FQ(tcut) c)

2 4 6 8

J tcut

10

6

10

4

10

2

FQ FQ(tcut) FQ

d)

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Model Overview

Two fermionic species with on-site interactions in a 1D chain: H0 = −J

  • x,σ
  • c†

σxcσx+1 + h.c.

  • + U
  • x
  • c†

↓xc↓xc† ↑xc↑x

  • Staggered magnetization/density:

O± =

  • x

(−1)x c†

↑xc↑x ∓ c† ↓xc↓x

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Entanglement Certified with O±

15 10 5 5 10 15 U/J 0.2 0.4 0.6 0.8 1.0 T/J 10 20 30 40 50 60

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Outlook

◮ Implementation in ultra-cold atoms experiments ◮ Generalize to different ensembles ◮ Analogous results for local thermalization/ETH ◮ Probe entanglement in topological states of matter

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Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model

Thank you