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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

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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,


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


Multipartite entanglement

  • Multipartite entanglement is complicated!

  • - Operational: no good single “unit” of entanglement

  • - Abstract: inequivalent ordering of states

  • Many different KINDS of entanglement


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


SEP

Distance-like entanglement measures

  • “Distance” to closest separable state


SEP

Distance-like entanglement measures

  • “Distance” to closest separable state

  • Different interpretations


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


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


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


    Geometric Measure

    Relative entropy of entanglement

    Logarithmic Robustness

    Equivalence of measures

    • When does equality hold?


    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


    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


    Geometric Measure

    Relative entropy of entanglement

    Logarithmic Robustness

    Equivalence of measures

    • Average over local to get

    • where are projections onto invariant subspace


    Geometric Measure

    Relative entropy of entanglement

    Logarithmic Robustness

    Equivalence of measures

    • Average over local to get

    • where are projections onto invariant subspace

    • Valid candidate?

    ?


    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 :

    ?


    Equivalence of measures

    • Equivalence is given by

    • Find local group such that

    • Found for - Stabilizer states

      • - Symmetric basis states

      • - Anti-symmetric basis states


    Stabilizer States

    • qubits

    • “Common eigen-state of stabilizer group

      • .”

    Commuting Pauli operators


    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)


    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)


    Stabilizer States

    • Average over stabilizer group

    • Don’t need to know

    • For all stabilizer states

    where

    for any generators


    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).


    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).


    Permutation symmetric basis states

    • Average over

    • For symmetric basis states


    Relationship to entanglement witnesses

    • Entanglement Witness

    SEP


    Relationship to entanglement witnesses

    • Entanglement Witness

    • Geometric measure

    SEP


    Relationship to entanglement witnesses

    • Entanglement Witness

    • Geometric measure

    • Robustness

    SEP


    Relationship to entanglement witnesses

    • Entanglement Witness

    • Geometric measure

    • Robustness

    • Optimality of

    • - can be shown that equivalence is a -OEW

    SEP


    ?

    ?

    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)


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