Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p^a-1) MUMEBs in \cd by properties of Guss sums for arbitrary odd d. It improves the known lower bound p^a-1 for odd d. Certainly, it also generalizes the lower bound 2(p^a-1) for d being a single prime power. Furthermore, we construct MUMEBs in \ckd for general k\geq 2 and odd d. We get the similar lower bounds as k,b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in \ckd, and obtain greater lower bounds than reducing the problem into prime power dimension in some cases.

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Corrigendum to “Minimum polynomials and lower bounds for eigenvalue multiplicities of prime-power order elements in representations of classical groups” [J. Algebra 243 (2001) 228–263]

Journal of Algebra ◽

10.1016/j.jalgebra.2005.02.006 ◽

2006 ◽

Vol 296 (1) ◽

pp. 249-252 ◽

Cited By ~ 2

Author(s):

L. Di Martino ◽

A.E. Zalesskii

Keyword(s):

Lower Bounds ◽

Prime Power ◽

Prime Power Order ◽

Classical Groups ◽

Eigenvalue Multiplicities

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On Fermat's equation with prime power exponents

Acta Arithmetica ◽

10.4064/aa-59-1-83-86 ◽

1991 ◽

Vol 59 (1) ◽

pp. 83-86

Author(s):

Cui-Xiang Zhong

Keyword(s):

Prime Power ◽

Fermat's Equation

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Minimum Polynomials and Lower Bounds for Eigenvalue Multiplicities of Prime-Power Order Elements in Representations of Classical Groups

Journal of Algebra ◽

10.1006/jabr.2001.8837 ◽

2001 ◽

Vol 243 (1) ◽

pp. 228-263 ◽

Cited By ~ 13

Author(s):

L Di Martino ◽

A.E Zalesskii

Keyword(s):

Lower Bounds ◽

Prime Power ◽

Prime Power Order ◽

Classical Groups ◽

Eigenvalue Multiplicities

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A new construction of mutually unbiased maximally entangled bases in ℂq⊗ℂq

International Journal of Quantum Information ◽

10.1142/s0219749921500143 ◽

2021 ◽

pp. 2150014

Author(s):

Dengming Xu ◽

Feiya Li ◽

Wei Hu

Keyword(s):

Lower Bounds ◽

Prime Power ◽

Maximal Size ◽

New Construction

This paper is devoted to constructing mutually unbiased maximally entangled bases (MUMEBs) in [Formula: see text], where [Formula: see text] is a prime power. We prove that [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd, where [Formula: see text] is the maximal size of the sets of MUMEBs in [Formula: see text]. This highly raises the lower bounds of [Formula: see text] given in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175. It should de noted that the method used in the paper is completely different from that in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175.

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New construction of mutually unbiased bases in square dimensions

Quantum Information and Computation ◽

10.26421/qic5.2-1 ◽

2005 ◽

Vol 5 (2) ◽

pp. 93-101

Author(s):

P. Wocjan ◽

T. Beth

Keyword(s):

Lower Bounds ◽

Prime Power ◽

Hadamard Matrices ◽

Latin Squares ◽

Mutually Unbiased Bases ◽

Asymptotic Growth ◽

Orthogonal Latin Squares ◽

Mutually Orthogonal Latin Squares ◽

New Construction

We show that k=w+2 mutually unbiased bases can be constructed in any square dimension d=s^2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the design-theoretic objects (s,k)-nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bounds on the asymptotic growth of the number of mutually orthogonal Latin squares (based on number theoretic sieving techniques), we obtain that the number of mutually unbiased bases in dimensions d=s^2 is greater than s^{1/14.8} for all s but finitely many exceptions. Furthermore, our construction gives more mutually unbiased bases in many non-prime-power dimensions than the construction that reduces the problem to prime power dimensions.

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