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A Quantum t-design is a probability distribution over pure quantum states which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree t or less. Specifically, the average of any polynomial function of degree t over the design is exactly the same as the average over Haar measure. Here the Haar measure is a uniform probability distribution over all quantum states. These designs are usually unique, and thus almost always calculable. Two particularly important types of t-designs in quantum mechanics are spherical and unitary t-designs. Spherical t-designs are designs where points of the design (i.e. the points being used for the averaging process) are points on a unit sphere. Spherical t-designs and variations thereof have been considered lately and found useful in quantum information theory,〔A. Hayashi, T. Hashimoto, M. Horibe. Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A, 72: 032325, 2006. Also quant-ph/0410207.〕 quantum cryptography and other related fields. Unitary designs are analogous to spherical designs in that they approximate the entire unitary group via a finite collection of unitary matrices. Unitary designs have been found useful in information theory〔C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs: constructions and applications, (2006).〕 and quantum computing. Unitary designs are especially useful in quantum computing since most operations are represented by unitary operators. == Motivation == In a d-dimensional Hilbert space when averaging over all quantum pure states the natural group is SU(d), the special unitary group of dimension d. The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries. A particularly widely used example of this is the spin system. For this system the relevant group is SU(2) which is the group of all 2x2 unitary operators. Since every 2x2 unitary operator is a rotation of the Bloch sphere, the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is ''the'' rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere. Another recent application is the fact that a symmetric informationally complete POVM is also a spherical 2-design. Also, since a 2-design must have more than elements, a SIC-POVM is a minimal 2-design. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quantum t-design」の詳細全文を読む スポンサード リンク
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