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# Error Analysis of a Numerical Calculation about One-Qubit Quantum Channel Capacity

Error Analysis of a Numerical Calculation about One-Qubit Quantum Channel Capacity . Kimikazu Kato 1,2 , Hiroshi Imai 1,3 , and Keiko Imai 4 1 Department of Computer Science, University of Tokyo 2 Nihon Unisys, Ltd. 3 ERATO-SORST Quantum Computation and Information

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## Error Analysis of a Numerical Calculation about One-Qubit Quantum Channel Capacity

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1. 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

2. 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]

3. Quantum Channel and Its Capacity Quantum state (continuous) Quantum state Quantum channel photon Encode noise Decode Message to send (discrete) 10010111000101100 0000010010010・・・・ Received Message 10010111000101100 0000010010010・・・・ Mathematically a channel is represented by an affine map

4. 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 state Mixed state pure states Appears on the boundary of the convex object mixed states

5. 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

6. 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.

7. 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. NOTE: Four is the maximum number of points to determine a ball in three dimensional space In Euclidean space, the smallest enclosing ball of the ellipsoid is determined by only two points The space is really distorted

8. 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]

9. We have proved the following facts: ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✖ : not defined ✔, ✔ : equivalent to the divergence-Voronoi [Kato et al. 05] [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

10. Numerical Calculation of Holevo Capacity for one-qubit [Oto, Imai, Imai ’04] Idea of the calculation: take some point and think of their image Calculate the SEB of the image w.r.t. a divergence Plot uniformly distributed points • A VD is used in its process • The coincidence of VD’s guarantees its effectiveness.

11. 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

12. Error bound General case: Not good bound Becomes very rapidly as Meaningless when and it is possible This should be improved

13. 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

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