On a Geometric Structure of Pure Multi-qubit Quantum States and Its Applicability to a Numerical Computation 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, 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. Generally for a numerical calculation of a continuous geometric object, it is natural to think of its approximation by discrete object

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

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

Why use Voronoi Diagram? A VD can be a effective indication to know how similar some distances are For a numerical approximation over a continuous object, it is natural to introduce a discrete structure to it Actually in a numerical calculation of Holevo capacity [Oto, Imai, Imai ’04], a Voronoi diagram is used. Hopefully there might be another application in quantum information theory

A density matrix represents a quantum state. 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”

Pure states Mixed states Stands for a state of a single particle States except pure states Stands for a probabilistic mixture of multiple particles This is equivalent to This is equivalent to

For 3 or higher level, the coincidence of VDs doesn’t occur Summary of Our Results For one-qubit pure quantum states, some VDs w.r.t. some distances coincide For 3 or higher level, the coincidence of VDs doesn’t occur ・・・ divergence Euclidean distance Fubini-Study distance [Kato, Oto, Imai, Imai ’05] Presented at Int. Symp. on VD 2005 divergence Euclidean distance New Result!

Parameterization of a Density Matrix One-qubit (2 level) case This is a ball; called Bloch ball Pure states appear on the surface of the ball d level case ( ) Generally very complicated inequities Complicated structure (not so simple as in one-qubit)

Distances of Pure Quantum States Fubini-Study distance Bures distance （Both are only defined for a pure state）

Divergence Classical version (In statistics): Kullback-Leibler divergence Distance between two probabilistic distributions Quantum version: Quantum divergence Where when Hence this cannot be defined when some eigenvalues are zero. Especially not defined for pure states. Note: although this indicates how near two states are, it doesn’t satisfy the axiom of distance.

Theorem 1 [Kato, Oto, Imai, Imai ’05] For one-qubit pure states, Voronoi diagrams with respect to the following distances are all the same Fubini-Study distance Bures distance Euclidean distance Geodesic distance Divergence Note: a divergence is not define for a pure state, but its diagram is defined taking a limit.

Numerical Calculation of Holevo Capacity [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 Calculate the SEB of the image w.r.t. a divergence Plot uniformly distributed points 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 (Theorem 1) 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.

Main Result For 3 or higher level, the VDs w.r.t. Euclidean distance and the divergence no longer coincide. Theorem 2 Some section of the space of quantum states with a hyper-plain is a solid ellipsoid In the section, pure states appear on the surface of the ellipsoid For the pure states, the VD w.r.t the divergence is the same as the geodesic VD when mapped to a sphere by an affine transform Mapped to a sphere by an affine transform VD for the section and pure w.r.t. the divergence Coincide with the VD w.r.t. geodesic distance

The Idea of Proof Whole space of quantum states: difficult to understand ？？ Because we don’t want to take log of this: Take a section with a hyper-plain. If a good hyper-plain is chosen, there appears a simple geometric structure. Now the calculation is much simpler

Examples Note: Example 1 in the paper is wrong Correctly Coincide!! VD w.r.t. Divergence VD w.r.t. Euclidean distance Example 3 Don’t coincide VD w.r.t. Divergence VD w.r.t. Euclidean distance

The coincidence above doesn’t occur for higher level. Conclusion For pure one-qubit states, Voronoi diagrams with respect to some distances coincide. The coincidence above doesn’t occur for higher level. ・・・

A program to draw this diagram is downloadable from my home page Appendix A program to draw this diagram is downloadable from my home page Visit http://www-imai.is.s.u-tokyo.ac.jp/~kkato/ or Search by Google for “Kimikazu Kato” But it uses a very naïve algorithm, suggestion for improvement is welcomed

Thank you ありがとうございました 감사합니다 谢谢