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

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

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

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

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

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Quantum Phase Estimation Goal: estimate a parameter θ =?

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 − i θ O ρ 0 e + i θ O

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 − i θ O ρ 0 e + i θ O How precise can this estimation be?

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Cram´ er-Rao Bound

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Cram´ er-Rao Bound Conditional probability distribution: f ( µ | θ )

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 θ = ˆ ˆ θ ( µ )

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 θ = ˆ ˆ θ ( µ ) Fisher information: � F = f ( µ | θ ) ( ∂ θ ln f ( µ | θ )) µ

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

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Quantum Cram´ er-Rao Bound

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 − i θ O ρ 0 e + i θ O

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 − i θ O ρ 0 e + i θ O Given some measurement setup: POVM { E µ }

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 − i θ O ρ 0 e + i θ O Given some measurement setup: POVM { E µ } ⇒ f ( µ | θ ) = Tr ( ρ ( θ ) E µ )

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 − i θ O ρ 0 e + i θ O Given some measurement setup: POVM { E µ } ⇒ f ( µ | θ ) = Tr ( ρ ( θ ) E µ ) ⇒ F ( { E µ } )

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 − i θ O ρ 0 e + i θ O Given some measurement setup: POVM { E µ } ⇒ f ( µ | θ ) = Tr ( ρ ( θ ) E µ ) ⇒ F ( { E µ } ) Quantum Fisher information: F Q = max { E µ } F ( { E µ } )

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 − i θ O ρ 0 e + i θ O Given some measurement setup: POVM { E µ } ⇒ f ( µ | θ ) = Tr ( ρ ( θ ) E µ ) ⇒ F ( { E µ } ) Quantum Fisher information: F Q = max { E µ } F ( { E µ } ) Best precision achievable with ρ 0 : Var (ˆ θ ) ≥ F − 1 Q

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model How to Calculate the QFI?

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model How to Calculate the QFI? Pure states ρ 0 = | ψ � � ψ | :

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model How to Calculate the QFI? Pure states ρ 0 = | ψ � � ψ | : � � ψ | O 2 | ψ � − � ψ | O | ψ � 2 � F Q = 4Var ( O , ψ ) = 4

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model How to Calculate the QFI? Pure states ρ 0 = | ψ � � ψ | : � � ψ | O 2 | ψ � − � ψ | O | ψ � 2 � F Q = 4Var ( O , ψ ) = 4 Mixed states ρ 0 = � λ ρ λ | λ � � λ | :

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model How to Calculate the QFI? Pure states ρ 0 = | ψ � � ψ | : � � ψ | O 2 | ψ � − � ψ | O | ψ � 2 � F Q = 4Var ( O , ψ ) = 4 Mixed states ρ 0 = � λ ρ λ | λ � � λ | : ρ λ − ρ λ ′ ρ λ + ρ λ ′ ( ρ λ − ρ λ ′ ) | � λ | O | λ ′ � | 2 , � F Q = 2 λ,λ ′

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

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

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Multipartite Entanglement System of N spins 1/2 | ψ � ∈ H 1 ⊗ · · · ⊗ H N

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Multipartite Entanglement System of N spins 1/2 | ψ � ∈ H 1 ⊗ · · · ⊗ H N Product states: | ψ � = | φ 1 � ⊗ · · · ⊗ | φ N �

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Multipartite Entanglement System of N spins 1/2 | ψ � ∈ H 1 ⊗ · · · ⊗ H N Product states: | ψ � = | φ 1 � ⊗ · · · ⊗ | φ N � k-producible states: | ψ � = | ψ i 1 � ⊗ · · · ⊗ | ψ i P � where each | ψ i p � is a state of at most k spins

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model Multipartite Entanglement System of N spins 1/2 | ψ � ∈ H 1 ⊗ · · · ⊗ H N Product states: | ψ � = | φ 1 � ⊗ · · · ⊗ | φ N � k-producible states: | ψ � = | ψ i 1 � ⊗ · · · ⊗ | ψ i P � where each | ψ i p � is a state of at most k spins ◮ Entangled states � = product states ◮ k-partite entangled states � = k-producible states

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model QFI as an Entanglement Witness

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model QFI as an Entanglement Witness For a k-producible state | ψ � = | ψ i 1 � ⊗ · · · ⊗ | ψ i P � and O = � j O j : � Var ( O , ψ ) = Var ( O i p , ψ i p ) i p

Background in Quantum Metrology Multipartite Entanglement Certification Quench Dynamics 1D Fermi-Hubbard Model QFI as an Entanglement Witness For a k-producible state | ψ � = | ψ i 1 � ⊗ · · · ⊗ | ψ i P � and O = � j O j : � Var ( O , ψ ) = Var ( O i p , ψ i p ) i p This leads to bounds for the F Q of k-producible states: F Q ≤ kN for O = 1 � σ z j 2 j