1. On a Geometric Structure of Pure Multi-qubit Quantum States and Its Applicability to a Numerical Computation
1,2Kimikazu Kato, 3Mayumi Oto,
1,4Hiroshi Imai, and 5Keiko Imai
1 Department of Computer Science, Univ. of Tokyo
2 Nihon Unisys, Ltd.
3 Toshiba Corporation
4 ERATO-SORST Quantum Computation and Information
5 Department of Information and System Engineering,
Chuo Univ.

2. Objective of Our Research
Using Voronoi diagrams,
Understand the structure of the space of quantum states, and
Clarify the relations among the distances defined in the space of quantum states
Why?
This could be a fundamental research toward estimating a capacity of a quantum communication channel.
Generally for a numerical calculation of a continuous geometric object, it is natural to think of its approximation by discrete object

3. Quantum Channel and Its Capacity
Quantum state
(continuous)
Quantum state
Quantum channel
photon
noise
Decode
Code
Received Message
・・・・
Message to send
(discrete)
・・・・
How much information can be sent via this channel?
Generally its calculation is difficult

4. Spaces and Distances Euclidean space Space of quantum states
Associated distances
Euclidean space
Euclidean distance
There is a natural embedding
Divergence
Space of quantum states
dimensional concave object
What is this structure?
Space of pure quantum states
Geodesic distance
Fubini-Study distance
Bures distance
dimensional hyper-surface
How related?

5. Why use Voronoi Diagram?
A VD can be a effective indication to know how similar some distances are
For a numerical approximation over a continuous object, it is natural to introduce a discrete structure to it
Actually in a numerical calculation of Holevo capacity [Oto, Imai, Imai ’04], a Voronoi diagram is used.
Hopefully there might be another application in quantum information theory

6. A density matrix represents a quantum state.
Quantum States
A density matrix represents a quantum state.
A density matrix is a complex square matrix which satisfies the following conditions:
Hermitian
Positive semi-definite
Trace is one
When its size is dxd, it is called “d-level”

7. Pure states Mixed states Stands for a state of a single particle
States except pure states
Stands for a probabilistic mixture of multiple particles
This is equivalent to
This is equivalent to

8. For 3 or higher level, the coincidence of VDs doesn’t occur
Summary of Our Results
For one-qubit pure quantum states, some VDs w.r.t. some distances coincide
For 3 or higher level, the coincidence of VDs doesn’t occur
・・・
divergence
Euclidean distance
Fubini-Study distance
[Kato, Oto, Imai, Imai ’05] Presented at Int. Symp. on VD 2005
divergence
Euclidean distance
New Result!

9. Parameterization of a Density Matrix
One-qubit (2 level) case
This is a ball; called Bloch ball
Pure states appear on the surface of the ball
d level case ( )
Generally very complicated inequities
Complicated structure (not so simple as in one-qubit)

10. Distances of Pure Quantum States
Fubini-Study distance
Bures distance
（Both are only defined for a pure state）

11. Divergence Classical version (In statistics):
Kullback-Leibler divergence
Distance between two probabilistic distributions
Quantum version:
Quantum divergence
Where
when
Hence this cannot be defined when some eigenvalues are zero.
Especially not defined for pure states.
Note: although this indicates how near two states are, it doesn’t satisfy the axiom of distance.

12. Theorem 1 [Kato, Oto, Imai, Imai ’05]
For one-qubit pure states, Voronoi diagrams with respect to the following distances are all the same
Fubini-Study distance
Bures distance
Euclidean distance
Geodesic distance
Divergence
Note: a divergence is not define for a pure state, but its diagram is defined taking a limit.

13. Numerical Calculation of Holevo Capacity [Oto, Imai, Imai ’04]
Quantum channel is defined as an affine transform between spaces of quantum states.
Holevo capacity is defined as a radius of the smallest enclosing ball of the image of a given channel w.r.t. a divergence
The second argument is taken as the center of SEB
Idea of the calculation: take some point and think of their image
Calculate the SEB of the image
w.r.t. a divergence
Plot uniformly distributed points
Note: in fact, the SEB doesn’t appear like this. It is more distorted.
Actually it is proved the SEB is determined by four points [Hayashi et. al ‘04].

14. Why is it important? Because… A VD is used in its process
The coincidence of adjacencies of Euclidean distance and the divergence (Theorem 1) guarantees its effectiveness.
Remind the source points are plotted so that they are uniform in the meaning of Euclidean distance, while the SEB is taken in the meaning of the divergence.

15. Main Result
For 3 or higher level, the VDs w.r.t. Euclidean distance and the divergence no longer coincide.
Theorem 2
Some section of the space of quantum states with a hyper-plain is a solid ellipsoid
In the section, pure states appear on the surface of the ellipsoid
For the pure states, the VD w.r.t the divergence is the same as the geodesic VD when mapped to a sphere by an affine transform
Mapped to a sphere by an affine transform
VD for the section and pure
w.r.t. the divergence
Coincide with the VD w.r.t. geodesic distance

16. The Idea of Proof
Whole space of quantum states: difficult to understand ？？
Because we don’t want to take log of this:
Take a section with a hyper-plain.
If a good hyper-plain is chosen, there appears a simple geometric structure.
Now the calculation is much simpler

17. Examples Note: Example 1 in the paper is wrong Correctly Coincide!!
VD w.r.t.
Divergence
VD w.r.t.
Euclidean distance
Example 3
Don’t coincide
VD w.r.t.
Divergence
VD w.r.t.
Euclidean distance

18. The coincidence above doesn’t occur for higher level.
Conclusion
For pure one-qubit states, Voronoi diagrams with respect to some distances coincide.
The coincidence above doesn’t occur for higher level.
・・・

19. A program to draw this diagram is downloadable from my home page
Appendix
A program to draw this diagram is downloadable from my home page
Visit or
Search by Google for “Kimikazu Kato”
But it uses a very naïve algorithm, suggestion for improvement is welcomed

20. Thank you
ありがとうございました
감사합니다 谢谢