damian markham university of tokyo
Download
Skip this Video
Download Presentation
Damian Markham University of Tokyo

Loading in 2 Seconds...

play fullscreen
1 / 28

Damian Markham University of Tokyo - PowerPoint PPT Presentation


  • 99 Views
  • Uploaded on

Entanglement and Group Symmetries: Stabilizer, Symmetric and Anti-symmetric states. Damian Markham University of Tokyo. IIQCI September 2007, Kish Island, Iran. Collaborators: S. Virmani , M. Owari, M. Murao and M. Hayashi,. Why Bother?. Multipartite entanglement important in

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Damian Markham University of Tokyo' - tanuja


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
damian markham university of tokyo

Entanglement and Group Symmetries:Stabilizer, Symmetric and Anti-symmetric states

Damian Markham

University of Tokyo

IIQCI

September 2007, Kish Island, Iran

Collaborators: S. Virmani , M. Owari, M. Murao and M. Hayashi,

slide2

Why Bother?

  • Multipartite entanglement important in
  • - Quantum Information: MBQC
  • Error Corrn...
  • - Physics: Many-body physics?
  • Still MANY questions….. significance, role, usefulness…
  • Deepen our understanding of role and usefulness of entanglement in QI and many-body physics
slide3

Multipartite entanglement

  • Multipartite entanglement is complicated!
  • - Operational: no good single “unit” of entanglement
  • - Abstract: inequivalent ordering of states
  • Many different KINDS of entanglement
slide4

Multipartite entanglement

  • Multipartite entanglement is complicated!
  • - Operational: no good single “unit” of entanglement
  • - Abstract: inequivalent ordering of states
  • So we SIMPLIFY:
      • - Take simple class of distance-like measures
      • - Use symmetries to
  • Many different KINDS of entanglement
  • Show equivalence of measures
  • Calculate the entanglement
slide5

Geometric Measure

  • Relative entropy of entanglement
  • Logarithmic Robustness

SEP

Distance-like entanglement measures

  • “Distance” to closest separable state
slide6

Geometric Measure

  • Relative entropy of entanglement
  • Logarithmic Robustness

SEP

Distance-like entanglement measures

  • “Distance” to closest separable state
  • Different interpretations
slide7

Geometric Measure

  • Relative entropy of entanglement
  • Logarithmic Robustness

SEP

Distance-like entanglement measures

  • “Distance” to closest separable state
  • Different interpretations
  • Diff difficulty to calculate

difficulty

* M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, PRL 96 (2006) 040501

slide8

Geometric Measure

  • Relative entropy of entanglement
  • Logarithmic Robustness

SEP

Distance-like entanglement measures

  • “Distance” to closest separable state
  • Different interpretations
  • Diff difficulty to calculate

difficulty

  • Hierarchy or measures:*

* M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, PRL 96 (2006) 040501

slide9

Geometric Measure

  • Relative entropy of entanglement
  • Logarithmic Robustness

SEP

Distance-like entanglement measures

  • In this talk we:
  • Use symmetries to - prove equivalence for
          • i) stabilizer states
  • ii) symmetric basis states
  • iii) antisymmetric states
  • (operational conicidence, easier calcn)
  • - calculate the geometric measure
  • Example of operational meaning: optimal entanglement witness
  • “Distance” to closest separable state
  • Different interpretations
  • Diff difficulty to calculate

difficulty

  • Hierarchy or measures:*

* M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, PRL 96 (2006) 040501

slide10

Geometric Measure

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

  • When does equality hold?
slide11

Geometric Measure

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

  • When does equality hold?
  • Strategy:
  • Use to find good guess for
  • by symmetry:
  • averaging over local groups
slide12

Geometric Measure

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

  • When does equality hold?
  • Strategy:
  • Use to find good guess for
  • by symmetry:
  • averaging over local groups
slide13

Geometric Measure

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

  • Average over local to get
  • where are projections onto invariant subspace
slide14

Geometric Measure

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

  • Average over local to get
  • where are projections onto invariant subspace
  • Valid candidate?

?

slide15

Geometric Measure

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

  • Average over local to get
  • where are projections onto invariant subspace
  • Valid candidate?
  • By definition is separable
  • Equivalence if :

?

slide16

Equivalence of measures

  • Equivalence is given by
  • Find local group such that
  • Found for - Stabilizer states
          • - Symmetric basis states
          • - Anti-symmetric basis states
slide17

Stabilizer States

  • qubits
  • “Common eigen-state of stabilizer group
          • .”

Commuting Pauli operators

slide18

2

1

3

4

Stabilizer States

  • qubits
  • “Common eigen-state of stabilizer group
          • .”
  • e.g. Graph states

Commuting Pauli operators

  • GHZ states
  • Cluster states (MBQC)
  • CSS code states (Error Correction)
slide19

2

1

3

4

Stabilizer States

  • qubits
  • “Common eigen-state of stabilizer group
          • .”
  • e.g. Graph states
  • Associated weighted graph states good aprox. g.s. to high intern. Hamiltns*

Commuting Pauli operators

  • GHZ states
  • Cluster states (MBQC)
  • CSS code states (Error Correction)

* S. Anders, M.B. Plenio, W. DÄur, F. Verstraete and H.J. Briegel, Phys. Rev. Lett. 97, 107206 (2006)

slide20

Stabilizer States

  • Average over stabilizer group
  • Don’t need to know
  • For all stabilizer states

where

for any generators

slide21

Permutation symmetric basis states

  • qubits
  • Occur as ground states in some Hubbard models

* Wei et al PRA 68 (042307), 2003 (c.f. M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani in preparation).

slide22

Permutation symmetric basis states

  • qubits
  • Occur as ground states in some Hubbard models
  • Closest product state is also permutation symmetric*
  • Entanglement

* Wei et al PRA 68 (042307), 2003 (c.f. M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani in preparation).

slide23

Permutation symmetric basis states

  • Average over
  • For symmetric basis states
slide25

Relationship to entanglement witnesses

  • Entanglement Witness
  • Geometric measure

SEP

slide26

Relationship to entanglement witnesses

  • Entanglement Witness
  • Geometric measure
  • Robustness

SEP

slide27

Relationship to entanglement witnesses

  • Entanglement Witness
  • Geometric measure
  • Robustness
  • Optimality of
  • - can be shown that equivalence is a -OEW

SEP

slide28

?

?

Partial results* - Cluster

- Steane code

Stabilizer states

Conclusions

  • Use symmetries to – prove equivalence of measures
  • – calculate geometric measure
  • Interpretations coincide (e.g. entanglement witness, LOCC state discrimination)
  • Only need to calculate geometric measure
  • Next:
  • more relevance of equivalence? Maximum of “class”?
  • - other classes of states?

+ M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, quantu-ph/immanent

* D. Markham, A. Miyake and S. Virmani, N. J. Phys. 9, 194, (2007)

ad