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 Tokyo2 Nihon Unisys, Ltd.

3 Toshiba Corporation4 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 sp

ace of quantum states, and– Clarify the relations among the distances defi

ned 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 state

noise

Message to send(discrete)

100101110001011000000010010010・・・・

Code

Received Message10010111000101100

0000010010010・・・・

Decode

How much information can be sent via this channel?

Quantum channel

Generally its calculation is difficult

photon

Spaces and Distances

Space of quantum states

Space of pure quantum states

Euclidean distance

Euclidean space

DivergenceBures distance

Associated distances

dimensional convex object

12 d

22 d dimensional hyper-surface

Geodesic distanceFubini-Study distance

What is this structure?

How related?

There is a natural embedding

Voronoi DiagramFor a given set of points (called sites), the Voronoi diagram is defined as:

Roughly regions of the influence around each of sites

Strictly

Why do we use a Voronoi diagram?

Voronoi diagram with 4 sites with respect to Euclidean distance

Because…

It reflects a structure of a metric space, andIt changes a continuous geometric problem into a discrete problem

A distance used in a VD can be generalUsing 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 sa

tisfies the following conditions:– Hermitian– Positive semi-definite– Trace is one

• When its size is dxd, it is called “d-level”• Each state can be classified as pure or mixed

*

01Tr

Pure state Mixed state

1rank 2rank Appears on the boundary of the convex object

pure states

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

divergence Euclidean distance Fubini-Study distance

・・・

e.g. for one-qubit pure states, Voronoi diagrams on a Bloch sphere look like:

Bures-Voronoi

Fubini-Study-Voronoi

Euclidean Voronoi

Geodesic Voronoi

One-qubit(= 2-level)

Pure

Mixed

3 or higher level

Pure ?

Table of Coincidences to the Divergence-Voronoi

[Kato et al. ’05]

We have proved the following facts:

[Kato et al. ’06a]

✔: equivalent to the divergence-Voronoi✖: not equivalent to the divergence-Voronoi

✔ ✔ ✔ ✔

✖

NOTE: “Pure” or “mixed” means where the diagram is considered; Voronoi sites are always taken as pure states

✔

✔ ✔

✔

✔: our latest result

: not defined

Distances of Quantum States

Tr1),(B d

Tr),(cos FS dFubini-Study distance (only for pure states)

Bures distance (both for pure and mixed states)

Quantum divergence (for mixed states) )log(logTr|| D

XX

d

1*XX

d

log

loglog

1* whenWhere

NOTE: must have a full rank because log 0 is not definedEspecially the divergence is not defined for pure states.

The quantum divergence cannot be defined for pure states,

but…a Voronoi diagram w.r.t. the divergence C

AN be defined for the whole space

taking a limit of the diagram for mixed states

Take limit

Bures-Voronoi

Fubini-Study-Voronoi

Euclidean Voronoi

Geodesic Voronoi

One-qubit(= 2-level)

Pure

Mixed

3 or higher level

Pure ?

Table of Coincidences to the Divergence-Voronoi(again)

✔ ✔ ✔ ✔

✖

✔

✔ ✔

✔

What does this work for?

Numerical Calculation of Holevo Capacity for one-qubit [Oto, Imai, Imai ’04]

Holevo capacity is defined as a radius of the smallest enclosing ball of the image of a given channel w.r.t. a divergence

Idea of the calculation: take some point and think of their imagePlot uniformly distributed points Calculate the SEB of the image

w.r.t. a divergence

Quantum channel is defined as an affine transform between spaces of quantum states.

The second argument is taken as the center of SEB

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 V

oronoi diagrams w.r.t. some distances.• Our result gives a reinterpretation of the st

ructure 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