Bounds for Systems of Lines, and Jacobi Polynomials

Delsarte, P.; Goethals, J.-M.; Seidel, J.J.

doi:10.1016/b978-0-12-189420-7.50020-7

Cited by 123 publications

(232 citation statements)

References 6 publications

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“…Other types of tight designs, such as tight spherical designs or tight complex projective designs, play an important role in algebraic combinatorics [9] and quantum information [18]. They are less significant here as they appear to be very rare: numerical searches have not revealed any tight t-designs at all for t > 1.…”

Section: Corollary 17 If α < 1 and X Is A Unitary 1-design Such Thatmentioning

confidence: 99%

“…These bounds are called relative bounds, in contrast with the absolute bound in Theorem 9 that depends only on the number of distinct inner product values. The proof, which in some sense uses linear programming, mimics the relative bounds of Delsarte [8] for codes in association schemes and the bounds of Delsarte, Goethals, and Seidel [9] for spherical codes. …”

Section: Relative Boundsmentioning

confidence: 99%

“…In Section 3, we introduce unitary designs and their dual notion, unitary codes. We give lower bounds on the size of a design in terms of its strength and upper bounds on the size a code in terms of its degree; we call these bounds the absolute bounds, as they closely resemble the absolute bounds for spherical t-designs and codes found by Delsarte, Goethals, and Seidel [9,10]. In Section 4, we introduce zonal orthogonal polynomials for U(d), which are used in Section 5 to give more precise bounds when distances in a design or code are specified; these bounds are called relative bounds.…”

Section: Introductionmentioning

confidence: 99%

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Unitary designs and codes

Roy

Scott

2009

Des. Codes Cryptogr.

100

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A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code -a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values -and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary tdesign in U(d), and we catalogue some t-designs that arise from finite groups. *

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Section: Corollary 17 If α < 1 and X Is A Unitary 1-design Such Thatmentioning

confidence: 99%

Section: Relative Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unitary designs and codes

Roy

Scott

2009

Des. Codes Cryptogr.

100

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“…As an example of such a degenerate curve, consider the points 25) where α 1 = σ 3 , α 2 = σ 5 . The corresponding structural equations are…”

Section: Eight-dimensional Casementioning

confidence: 99%

“…This makes a deep connection with the notion of mutually unbiased bases (MUBs) [25,26], which were introduced as a central tool for quantum state reconstruction [27]. They also play a relevant role in a proper understanding of complementarity [28][29][30][31], in cryptographic protocols [32,33], and in quantum error correction codes [34,35].…”

Section: Introductionmentioning

confidence: 99%

Discrete phase-space structure of n-qubit mutually unbiased bases

et al. 2009

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We work out the phase-space structure for a system of n qubits. We replace the field of real numbers that label the axes of the continuous phase space by the finite field GF(2 n ) and investigate the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We provide a simple classification of such curves and study in detail the four-and eight-dimensional cases, analyzing also the effect of local transformations. In this way, we provide a comprehensive phase-space approach to the construction of mutually unbiased bases for n qubits.

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Complex Spherical Designs and Codes

Roy

Suda

2013

J of Combinatorial Designs

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Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper, we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these association schemes are nonsymmetric.

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Bounds for Systems of Lines, and Jacobi Polynomials

Cited by 123 publications

References 6 publications

Unitary designs and codes

Unitary designs and codes

Discrete phase-space structure of n-qubit mutually unbiased bases

Complex Spherical Designs and Codes

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