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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,

- 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 is complicated!
- - Operational: no good single “unit” of entanglement
- - Abstract: inequivalent ordering of states

- 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

- Many different KINDS of entanglement

- Show equivalence of measures
- Calculate the entanglement

- Relative entropy of entanglement
- Logarithmic Robustness

SEP

Distance-like entanglement measures

- “Distance” to closest separable state

- Relative entropy of entanglement
- Logarithmic Robustness

SEP

Distance-like entanglement measures

- “Distance” to closest separable state
- Different interpretations

- 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

- 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

- 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

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

- When does equality hold?

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

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

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

- Average over local to get
- where are projections onto invariant subspace

Relative entropy of entanglement

Logarithmic Robustness

Equivalence of measures

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

?

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

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)

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)

- 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
- 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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