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Error Analysis of a Numerical
Calculation about One-Qubit
Quantum Channel Capacity
Kimikazu Kato1,2, Hiroshi Imai1,3,
and Keiko Imai4
1 Department of Computer Science, University of Tokyo
2 Nihon Unisys, Ltd.
3 ERATO-SORST Quantum Computation and Information
4 Department of Information and System Engineering, Chuo University
Importance of numerical computation
for quantum states


How much information can be sent via a
quantum channel is important. Now a
quantum channel and quantum cryptography
are not far from reality!
Explicit numerical computation gives a clue to
theoretical research.

For example, there is a numerical verification (not
proof) of the additivity conjecture
[Osawa and Nagaoka ’00]
Quantum Channel and Its Capacity
Quantum state
(continuous)
Quantum channel
Quantum state
photon
Encode
Message to send
(discrete)
10010111000101100
0000010010010・・・・
noise
Decode
Received Message
10010111000101100
0000010010010・・・・
Mathematically a channel is represented by an affine map
Quantum States




A density matrix represents a quantum state.
A density matrix  is a complex square matrix which satisfies
the following conditions:
  *
 Hermitian
 0
 Positive semi-definite
Tr   1
 Trace is one
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
Geometry for one-qubit states



Any state can be represented by a point in a
ball (called “Bloch ball”)
A pure state corresponds to a point on a
sphere
Thus the image of a channel is an ellipsoid
Holevo capacity


Holevo capacity of a given channel is defined
as a radius of the smallest enclosing ball
which contains the image of the channel.
The radius used here is not Euclidean
distance, but the quantum divergence,
informational distance between two points.


The ball looks very distorted, and far from
intuition
Actually, there is an ellipsoid whose smallest
enclosing ball is determined by four points
[Hayashi et al ’04]
The smallest enclosing ball w.r.t. the
quantum divergence can be
determined by four points.
In Euclidean space, the smallest enclosing ball
of the ellipsoid is determined by only two points
NOTE: Four is the maximum number of points to
determine a ball in three dimensional space
The space is really distorted
Related work

We mathematically proved the coincidence of
Voronoi diagrams w.r.t some (pseudo-)
distances for one-qubit states.
[Kato, Oto, Imai, Imai ‘05, Kato, Oto, Imai, Imai ‘06b]

We also showed that for higher level pure
states, Voronoi diagrams w.r.t some
distances also coincide, but Euclidean
Voronoi diagram does not coincide with them.
[Kato, Oto, Imai, Imai ’06a]
We have proved the following facts:
BuresVoronoi
One-qubit
(= 2-level)
3 or higher
level
Pure
✔
Mixed
✔
Pure
✔
✔,
[Kato et al. 05]
✔
Fubini-StudyVoronoi
✔
Euclidean
Voronoi
✔
Geodesic
Voronoi
✔
✔
✔
✖
?
: not defined
: equivalent to the divergence-Voronoi
[Kato et al. 06b]
✖
: not equivalent to the divergence-Voronoi
[Kato et al. 06a]
NOTE: “Pure” or “mixed” means where the diagram is considered; Voronoi sites are always
taken as pure states
Numerical Calculation of Holevo Capacity
for one-qubit [Oto, Imai, Imai ’04]
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
 A VD is used in its process
 The coincidence of VD’s guarantees its effectiveness.
Importance of the error analysis

Knowledge about the error makes the actual
computation fast.


It tells when to stop the converging process
Faster computation makes more experiments
possible

It might enable experiments about exhaustive
number of samples in quantum state space; it will
be a strong support for a theoretical research
Error bound
General case:
Not good bound
Becomes very rapidly as
Meaningless when
and it is possible
This should be improved
Conclusion

We showed an explicit upper bound for the
error of the computation of one-qubit channel
capacity
Future work


Error bound for higher level system
Effective algorithm and actual computation
for higher level system