Geometry of the threequbit state, entanglement and division algebras

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Geometry of the three-qubit state, entanglement
and division algebras
Bogdan A. Bernevig and Handong Chen Stanford
University
J. Phys. A Math. Gen. 36 8325 (2003) (quant-ph/03
02081)
APS March Meeting Montreal, March 24 2004
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Motivation
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1-qubit state and 1st Hopf fibration

• The pure one-qubit state can be represented as

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2-qubit state and 2nd Hopf fibration

• Introduce quaternions q1 and q2

R. Mosseri and R. Dandoloff, J. Phys. A Math.
Gen. 34, 10243 (2001).
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2-qubit state and 2nd Hopf fibration

• Partial traced density matrix

R. Mosseri and R. Dandoloff, J. Phys. A Math.
Gen. 34, 10243 (2001).
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2-qubit state and 2nd Hopf fibration

• Partial traced density matrix

R. Mosseri and R. Dandoloff, J. Phys. A Math.
Gen. 34, 10243 (2001).
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3-qubit state and 3rd Hopf fibration

• Introduce octonions o1 and o2

B.A. Bernevig and H.D. Chen, J. Phys. A Math.
Gen. 36, 8325 (2003).
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3-qubit state and 3rd Hopf fibration

• Partial traced density matrix

B.A. Bernevig and H.D. Chen, J. Phys. A Math.
Gen. 36, 8325 (2003).
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3-qubit state and 3rd Hopf fibration

• Partial traced density matrix

B.A. Bernevig and H.D. Chen, J. Phys. A Math.
Gen. 36, 8325 (2003).
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Conclusion

• We reviewed the deep relation between 1-qubit (2
  qubit ) and the first (second) Hopf map. A
  entanglement sensitive 2nd Hopf map can be
  constructed.
• We show 3-qubit state is also related to the
  last, 3rd Hopf map. A entanglement sensitive map
  is explicitly constructed.

For more technique details, please refer to J.
Phys. A Math. Gen. 36 8325 (2003) or
quant-ph/0302081 The algebra of quaternion and
octonion is nicely reviewed by John C. Baez in
math.RA/0105155