Title: Voronoi Diagrams for Pure 1-qubit Quantum States
1Voronoi 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.
2Goal 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
3Basics 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
4Pure 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
5Definitions
- 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
6Distances and Divergences
- Bures distance in pure states
- Fubini-Study distance in pure states
- Divergences (only defined in mixed states)
- Holevo Capacity
7Computation 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
8Coincidence 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)
9Voronoi 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)
10Extension 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?
11Coincidence 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
12Conclusion
- 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
13Future 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.