8 research outputs found

    Mutually unbiased maximally entangled bases from difference matrices

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    Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish qq mutually unbiased bases with qβˆ’1q-1 maximally entangled bases and one product basis in CqβŠ—Cq\mathbb{C}^q\otimes \mathbb{C}^q for arbitrary prime power qq. In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in C12βŠ—C12\mathbb{C}^{12}\otimes \mathbb{C}^{12} and C21βŠ—C21\mathbb{C}^{21}\otimes\mathbb{C}^{21}, which improve the known lower bounds for d=3md=3m, with (3,m)=1(3,m)=1 in CdβŠ—Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}. Furthermore, we construct p+1p+1 mutually unbiased bases with pp maximally entangled bases and one product basis in CpβŠ—Cp2\mathbb{C}^p\otimes \mathbb{C}^{p^2} for arbitrary prime number pp.Comment: 24 page

    Two Problems of Gerhard Ringel

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    Gerhard Ringel was an Austrian Mathematician, and is regarded as one of the most influential graph theorists of the twentieth century. This work deals with two problems that arose from Ringel\u27s research: the Hamilton-Waterloo Problem, and the problem of R-Sequences. The Hamilton-Waterloo Problem (HWP) in the case of Cm-factors and Cn-factors asks whether Kv, where v is odd (or Kv-F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. Chapter 1 gives some general constructions for such decompositions and apply them to the case where m=3 and n=3x. This problem is settle for odd v, except for a finite number of x values. When v is even, significant progress is made on the problem, although open cases are left. In particular, the difficult case of v even and s=1 is left open for many situations. Chapter 2 generalizes the Hamilton-Waterloo Problem to complete equipartite graphs K(n:m) and shows that K(xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s,r≠1, gcd(x,z)=gcd(y,z)=1 and xyz≠0 (mod 4). Some more general constructions are given for the case when the cycles in a given two factor may have different lengths. These constructions are used to find solutions to the Hamilton-Waterloo problem for complete graphs. Chapter 3 completes the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups

    Master index of volumes 161–170

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    Part I:

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    Subject Index Volumes 1–200

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