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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

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Damian Markham University of Tokyo

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  1. 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,

  2. 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

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

  4. 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

  5. Geometric Measure • Relative entropy of entanglement • Logarithmic Robustness SEP Distance-like entanglement measures • “Distance” to closest separable state

  6. Geometric Measure • Relative entropy of entanglement • Logarithmic Robustness SEP Distance-like entanglement measures • “Distance” to closest separable state • Different interpretations

  7. 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

  8. 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

  9. 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

  10. Geometric Measure Relative entropy of entanglement Logarithmic Robustness Equivalence of measures • When does equality hold?

  11. 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

  12. 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

  13. Geometric Measure Relative entropy of entanglement Logarithmic Robustness Equivalence of measures • Average over local to get • where are projections onto invariant subspace

  14. Geometric Measure Relative entropy of entanglement Logarithmic Robustness Equivalence of measures • Average over local to get • where are projections onto invariant subspace • Valid candidate? ?

  15. 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 : ?

  16. Equivalence of measures • Equivalence is given by • Find local group such that • Found for - Stabilizer states • - Symmetric basis states • - Anti-symmetric basis states

  17. Stabilizer States • qubits • “Common eigen-state of stabilizer group • .” Commuting Pauli operators

  18. 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)

  19. 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)

  20. Stabilizer States • Average over stabilizer group • Don’t need to know • For all stabilizer states where for any generators

  21. 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).

  22. 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).

  23. Permutation symmetric basis states • Average over • For symmetric basis states

  24. Relationship to entanglement witnesses • Entanglement Witness SEP

  25. Relationship to entanglement witnesses • Entanglement Witness • Geometric measure SEP

  26. Relationship to entanglement witnesses • Entanglement Witness • Geometric measure • Robustness SEP

  27. Relationship to entanglement witnesses • Entanglement Witness • Geometric measure • Robustness • Optimality of • - can be shown that equivalence is a -OEW SEP

  28. ? ? 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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