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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