1. Voronoi Diagrams and a Numerical Estimation of a Quantum Channel Capacity 1,2 Kimikazu Kato, 3 Mayumi Oto, 1,4 Hiroshi Imai, and 5 Keiko 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, –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.

3. Quantum Channel and Its Capacity Quantum state (continuous) Quantum state noise Message to send (discrete) 10010111000101100 0000010010010 ・・・・ Code Received Message 10010111000101100 0000010010010 ・・・・ Decode How much information can be sent via this channel? Quantum channel Generally its calculation is difficult photon

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

5. 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 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, 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

6. 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” Each state can be classified as pure or mixed Pure stateMixed state Appears on the boundary of the convex object pure states mixed states

7. 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 divergenceEuclidean distanceFubini-Study distance ・・・ e.g. for one-qubit pure states, Voronoi diagrams on a Bloch sphere look like:

8. 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

9. Distances of Quantum States Fubini-Study distance (only for pure states) Bures distance (both for pure and mixed states) Quantum divergence (for mixed states) when Where NOTE: must have a full rank because log 0 is not defined Especially the divergence is not defined for pure states.

10. 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

11. 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?

12. 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 image Plot 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].

13. 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.

14. 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

15. Thank you