1. Voronoi Diagrams for Pure 1-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.

2. 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

3. 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

4. Pure State and 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

5. 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

6. Distances and Divergences
Bures distance in pure states
Fubini-Study distance in pure states
Divergences (only defined in mixed states)
Holevo Capacity

7. 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

8. 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 ordinary geodesic distance
The section of 3-dim Euclidean Voronoi diagram with the sphere
(Drawn with a tool made by K. Sugihara)

9. 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)

10. 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?

11. 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

12. 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

13. 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.