Voronoi Diagrams for Pure 1-qubit Quantum States

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Voronoi Diagrams for Pure1-qubit Quantum States

• Kimikazu Kato1,2, Mayumi Oto3,
• Hiroshi Imai1,4, and Keiko Imai5
• 1 Graduate School of Information Science and
  Technology, Univ. of Tokyo
• 2 Nihon Unisys, Ltd.
• 3 COE Super Robust Computation Project, Univ. of
  Tokyo
• 4 ERATO Quantum Computation and Information
• 5 Department of Information and System
  Engineering,
• Chuo Univ.

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Goal of our research

• Show some relations between some distances and
  divergences of quantum states
• Bures distance, Fubini-Study distance
• Primal divergence, dual divergence
• Use Voronoi diagrams as a tool
• To know about the structure of a metric space
• A good application is already known

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Basics of Quantum States

• A density matrix represents a quantum system
  which stands for an ensemble of particles
• A density matrix is a complex matrix which
  satisfies the followings
• Hermitian
• Positive semi-definite
• Tr 1
• Especially in 1-qubit case, the density matrix is
  2 x 2 and expressed as

Bloch ball
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Pure State and Mixed State
Pure state
Mixed state
Generally
Can be regarded as a probabilistic distribution
of pure states
where is a column vector and stands for
conjugate
Some of its eigenvalues are zero
All of its eigenvalues are non-zero
In 1-qubit case
Corresponds to a point in the interior of the ball
Corresponds to a point the surface of the ball
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Definitions

• Log of density matrix
• Quantum Channel
• Defined to be an affine transform from a state
  space to a state space

where
In 1-qubit case
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Distances and Divergences

• Bures distance in pure states
• Fubini-Study distance in pure states
• Divergences (only defined in mixed states)
• Holevo Capacity

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Computation of Holevo Capacity on 1-qubit States
(Oto-Imai-Imai 2004)

• Plot sufficiently many points
• Compute images of plotted points
• Work out the radius of smallest enclosing ball of
  image points
• Fathest Voronoi diagrams under the divergences
• Lower envelope

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Coincidence of Voronoi Diagrams under Some
Distances

• In the space of 1-qubit states, the followings
  are all equivalent
• The diagram under
• The diagram under
• The diagram under the ordinary geodesic distance
• The section of 3-dim Euclidean Voronoi diagram
  with the sphere

(Drawn with a tool made by K. Sugihara)
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Voronoi Diagrams under Divergences

• For a given set of points , two diagrams are
  defined as

has only planar edges
has non-planar edges
(proved in Oto-Imai-Imai 2004)
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Extension to Pure States

• Now it is natural to consider the coincidence of
  diagrams under divergences and distances.
• But unfortunately, the divergence
  can not be defined when is a pure state.

The eigenvalues of can be zero because
can be naturally defined as 0
But the eigenvalues of can NOT be zero.
Even so, does the Voronoi diagram converge in the
pure states?
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Coincidence of Voronoi Diagrams under Divergences
and Distances

• Voronoi diagrams under the divergences converge
  in pure states.
• The diagrams under the divergences coincide with
  the diagrams under the distances.

Take limit
Naturally extended to the pure states
Only defined in mixed states
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Conclusion

• In the 1-qubit pure states, followings are all
  equivalent
• The Voronoi diagrams under Fubini-Study, Bures,
  and geodesic distance
• The section of 3-dim Euclidean Voronoi diagram
  with the sphere
• The natural extensions of the Voronoi diagrams
  under the divergences

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Future Work

• Can this result be extended to a higher
  dimension?
• Our conjecture is No.
• What about other metrics?
• SLD Fisher metric, RLD Fisher metric, etc.