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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