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Voronoi Diagrams and a Numerical
Estimation of a Quantum Channel Capacity
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.
Objective of Our Research
Using Voronoi diagrams,
– We want to 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.
Quantum Channel and Its Capacity
Quantum state
(continuous)
Quantum channel
Quantum state
photon
noise
Code
Message to send
(discrete)
10010111000101100
0000010010010・・・・
Decode
Received Message
10010111000101100
0000010010010・・・・
How much information can be sent via this channel?
Generally its calculation is difficult
Spaces and Distances
Associated distances
Euclidean space
Euclidean distance
There is a natural
embedding
Space of
quantum states
d 2  1 dimensional
convex object

What is this
structure?
Divergence
Bures distance
Space of pure
quantum states
2d  2 dimensional
hyper-surface
Geodesic distance
Fubini-Study distance
How related?
Voronoi Diagram
For a given set of points
(called sites),
the Voronoi diagram is defined as:
Roughly
regions of the influence around each of sites
Strictly
Voronoi diagram with 4 sites with respect to
Euclidean distance
Why do we use a Voronoi diagram?
Because…
It reflects a structure of a metric space, and
It changes a continuous geometric problem into a discrete problem
A distance used in a VD can be general
Using VDs, we can compare some distances defined in a
quantum state space
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

 0
Tr   1
• When its size is dxd, it is called “d-level”
• Each state can be classified as pure or mixed
pure states
Pure state
Mixed state
rank   1
rank   2
Appears on the boundary
of the convex object
mixed states
Summary of Our Results
• We considered Voronoi diagrams when
sites are given as pure states, and
• Proved coincidences among Voronoi
diagrams w.r.t. some distances
e.g. for one-qubit pure states, Voronoi diagrams on a Bloch sphere look like:
・・・
divergence
Euclidean distance
Fubini-Study distance
Table of Coincidences to the Divergence-Voronoi
We have proved the following facts:
One-qubit
(= 2-level)
3 or higher
level
Pure
BuresVoronoi
FubiniStudyVoronoi
✔
✔
Euclidean
Voronoi
✔
Geodesic
Voronoi
✔
[Kato et al. ’05]
Mixed
✔
Pure
✔
✔
✔
✖
[Kato et al. ’06a]
?
: not defined
✔: equivalent to the divergence-Voronoi
✖: not equivalent to the divergence-Voronoi
✔: our latest result
NOTE: “Pure” or “mixed” means where the diagram is considered; Voronoi sites are always
taken as pure states
Distances of Quantum States
Quantum divergence (for mixed states)
D ||    Tr  (log   log  )
 log 1

 1





*
*

X


X

when
Where log   X 


X


log d 
d 


NOTE:  must have a full rank because log 0 is not defined
Especially the divergence is not defined for pure states.
Fubini-Study distance (only for pure states)
cos d FS (  ,  )  Tr 
Bures distance (both for pure and mixed states)
d B (  ,  )  1  Tr
 
The quantum divergence cannot be
defined for pure states,
but…
a Voronoi diagram w.r.t. the divergence
CAN be defined for the whole space
taking a limit of the diagram for
mixed states
Take limit
Table of Coincidences to the Divergence-Voronoi
(again)
One-qubit
(= 2-level)
3 or higher
level
BuresVoronoi
FubiniStudyVoronoi
Pure
✔
✔
Mixed
✔
Pure
✔
Euclidean
Voronoi
✔
Geodesic
Voronoi
✔
✔
✔
✖
What does this work for?
?
Numerical Calculation of Holevo Capacity for onequbit [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
Plot uniformly distributed points
Calculate the SEB of the image
w.r.t. a divergence
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].
Why is it important?
Because…
 A VD is used in its process
 The coincidence of adjacencies of Euclidean
distance and the divergence 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.
Conclusion
• We showed some coincidences among
Voronoi diagrams w.r.t. some distances.
• Our result gives a reinterpretation of the
structure of a quantum state space, and is
also useful for calculation of a quantum
channel capacity
Future work
• Numerical computation of a quantum
channel capacity for 3 or higher level
system
According to the theorem we showed, a naïve extension of the
method used for the one-qubit system is not effective
Thank you