Entanglement and Group Symmetries Stabilizer Symmetric and Antisymmetric
Entanglement and Group Symmetries: Stabilizer, Symmetric and Anti-symmetric states Damian Markham University of Tokyo Collaborators: S. Virmani , M. Owari, M. Murao and M. Hayashi, IIQCI September 2007, Kish Island, Iran
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! • Many different KINDS of entanglement - Operational: no good single “unit” of entanglement - Abstract: inequivalent ordering of states
Multipartite entanglement • Multipartite entanglement is complicated! • Many different KINDS of entanglement • - 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 (1) Show equivalence of measures (2) Calculate the entanglement
Distance-like entanglement measures • “Distance” to closest separable state • Geometric Measure • Relative entropy of entanglement SEP • Logarithmic Robustness
Distance-like entanglement measures • “Distance” to closest separable state SEP • • Geometric Measure • Relative entropy of entanglement Different interpretations • Logarithmic Robustness
Distance-like entanglement measures • “Distance” to closest separable state • Different interpretations • Diff difficulty to calculate difficulty SEP • Geometric Measure • Relative entropy of entanglement • Logarithmic Robustness * M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, PRL 96 (2006) 040501
Distance-like entanglement measures • “Distance” to closest separable state • Different interpretations • Diff difficulty to calculate • Hierarchy or measures: * difficulty SEP • Geometric Measure • Relative entropy of entanglement • Logarithmic Robustness * M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, PRL 96 (2006) 040501
Distance-like entanglement measures • In this talktowe: “Distance” closest separable state • Use symmetries to • Geometric Measure - prove equivalence for i) stabilizer states SEP difficulty ii) symmetric basis states • Relative entropy iii) antisymmetric statesof entanglement (operational conicidence, easier calcn) • Different interpretations • Diff difficulty to calculate • Hierarchy or measures: * - calculate the geometric measure • Example of operational meaning: optimal entanglement witness • Logarithmic Robustness * M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, PRL 96 (2006) 040501
Equivalence of measures • When does equality hold? Geometric Measure Relative entropy of entanglement Logarithmic Robustness
Equivalence of measures • When does equality hold? Geometric Measure Strategy: • Use to find good guess for Relative entropy of entanglement by symmetry: averaging over local groups Logarithmic Robustness
Equivalence of measures • When does equality hold? Geometric Measure Strategy: • Use to find good guess for Relative entropy of entanglement by symmetry: averaging over local groups Logarithmic Robustness
Equivalence of measures • Average over local to get Geometric Measure where are projections onto invariant subspace Relative entropy of entanglement Logarithmic Robustness
Equivalence of measures • Average over local to get Geometric Measure where are projections onto invariant subspace Relative entropy of entanglement • Valid candidate? ? Logarithmic Robustness
Equivalence of measures • Average over local to get Geometric Measure where are projections onto invariant subspace Relative entropy of entanglement • Valid candidate? ? • By definition • Equivalence if : is separable Logarithmic Robustness
Equivalence of measures Equivalence is given by • Find local group • Found for such that - Stabilizer states - Symmetric basis states - Anti-symmetric basis states
Stabilizer States • qubits • “Common eigen-state of stabilizer group. ” Commuting Pauli operators
Stabilizer States • qubits • “Common eigen-state of stabilizer group. ” Commuting Pauli operators • e. g. Graph states 2 1 3 4 - GHZ states - Cluster states (MBQC) - CSS code states (Error Correction)
Stabilizer States • qubits • “Common eigen-state of stabilizer group. ” Commuting Pauli operators • e. g. Graph states 2 1 3 - GHZ states - Cluster states (MBQC) - CSS code states (Error Correction) 4 • Associated weighted graph states good aprox. g. s. to high intern. Hamiltns* * 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 where for any • Don’t need to know • For all stabilizer states 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 SEP • Geometric measure • Robustness • Optimality of - can be shown that equivalence is a -OEW
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 Stabilizer states 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) ? Partial results* - Cluster - Steane code
- Slides: 28