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

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 Why Bother?

Multipartite entanglement is complicated! - Operational: no good single “unit” of entanglement - Abstract: inequivalent ordering of states Multipartite entanglement Many different KINDS of 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 Multipartite entanglement Many different KINDS of entanglement (1)Show equivalence of measures (2)Calculate the entanglement

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

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

Distance-like entanglement measures Geometric Measure Relative entropy of entanglement Logarithmic Robustness “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) SEP

Distance-like entanglement measures Geometric Measure Relative entropy of entanglement Logarithmic Robustness “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) SEP

Distance-like entanglement measures Geometric Measure Relative entropy of entanglement Logarithmic Robustness “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) SEP 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

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

Geometric Measure Relative entropy of entanglement Logarithmic Robustness When does equality hold? Strategy: Use to find good guess for by symmetry: averaging over local groups Equivalence of measures

When does equality hold? Strategy: Use to find good guess for by symmetry: averaging over local groups Equivalence of measures Geometric Measure Relative entropy of entanglement Logarithmic Robustness

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

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

Average over local to get where are projections onto invariant subspace Valid candidate? By definition is separable Equivalence if : Equivalence of measures ? Geometric Measure Relative entropy of entanglement Logarithmic Robustness

Equivalence is given by Find local groupsuch that Found for - Stabilizer states - Symmetric basis states - Anti-symmetric basis states Equivalence of measures

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

qubits “Common eigen-state of stabilizer group.” e.g. Graph states Commuting Pauli operators GHZ states - Cluster states (MBQC) - CSS code states (Error Correction) 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, (2006) Stabilizer States

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

qubits Occur as ground states in some Hubbard models Permutation symmetric basis states * Wei et al PRA 68 (042307), 2003 (c.f. M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani in preparation).

qubits Occur as ground states in some Hubbard models Closest product state is also permutation symmetric* Entanglement Permutation symmetric basis states * Wei et al PRA 68 (042307), 2003 (c.f. M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani in preparation).

Average over For symmetric basis states Permutation symmetric basis states

Entanglement Witness Relationship to entanglement witnesses SEP

Entanglement Witness Geometric measure Relationship to entanglement witnesses SEP

Entanglement Witness Geometric measure Robustness Relationship to entanglement witnesses SEP

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

Conclusions ? ? Stabilizer states Partial results* - Cluster - Steane code 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)