scholarly journals Block-circulant matrices with circulant blocks, Weil sums, and mutually unbiased bases. II. The prime power case

2009 ◽  
Vol 50 (3) ◽  
pp. 032104 ◽  
Author(s):  
Monique Combescure
Keyword(s):  
Prime Power ◽  
Circulant Matrices ◽  
Mutually Unbiased Bases ◽  
Weil Sums

New bounds of Mutually unbiased maximally entangled bases in C^dxC^(kd)

2018 ◽  
Vol 18 (13&14) ◽  
pp. 1152-1164
Author(s):  
Xiaoya Cheng ◽  
Yun Shang
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Power ◽  
Latin Squares ◽  
Mutually Unbiased Bases ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Bipartite System

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.

scholarly journals Covariant mutually unbiased bases

2016 ◽  
Vol 28 (04) ◽  
pp. 1650009 ◽  
Author(s):  
Claudio Carmeli ◽  
Jussi Schultz ◽  
Alessandro Toigo
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Equivalence Class ◽  
Symplectic Group ◽  
Prime Power ◽  
Mutually Unbiased Bases ◽  
Full Group ◽  
Unitary Operators ◽  
Oscillator Group ◽  
Dimension Formula

The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension [Formula: see text] covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.

New construction of mutually unbiased bases in square dimensions

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.

scholarly journals Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

2021 ◽  
Vol 29 (1) ◽  
pp. 15-34
Author(s):  
Daniel Uzcátegui Contreras ◽  
Dardo Goyeneche ◽  
Ondřej Turek ◽  
Zuzana Václavíková
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Circulant Matrix ◽  
Quantum Information Theory ◽  
Circulant Matrices ◽  
Mutually Unbiased Bases ◽  
Roots Of Unity ◽  
Finite Rings ◽  
Absolute Value ◽  
Important Open Problem

Abstract It is known that a real symmetric circulant matrix with diagonal entries d ≥ 0, off-diagonal entries ±1 and orthogonal rows exists only of order 2d + 2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d ≥ 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with d different from an odd integer is n = 2d + 2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings ℤ m . As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.

Deterministic quantum distribution of a d-ary key

2009 ◽  
Vol 9 (11&12) ◽  
pp. 50-62
Author(s):  
A. Eusebi ◽  
S. Mancini
Keyword(s):  
Explicit Expression ◽  
Quantum Key Distribution ◽  
Prime Power ◽  
Key Distribution ◽  
Mutually Unbiased Bases ◽  
Quantum Distribution ◽  
Quantum Key Distribution Protocol ◽  
Individual Attack

We present an extension to a d-ary alphabet of a recently proposed deterministic quantum key distribution protocol. It relies on the use of mutually unbiased bases in prime power dimension d, for which we provide an explicit expression. Then, by considering a powerful individual attack, we show that the security of the protocol is maximal for d=3.

Efficient 2-designs from bases exist

2008 ◽  
Vol 8 (8&9) ◽  
pp. 734-740
Author(s):  
G. McConnell ◽  
D. Gross
Keyword(s):  
Prime Power ◽  
Combinatorial Problem ◽  
Quantum Information Theory ◽  
Mutually Unbiased Bases ◽  
Dimensional Vector ◽  
Nonlinear Functions ◽  
Dimensional Vector Space ◽  
Highly Nonlinear ◽  
Efficient Alternative ◽  
State Tomography

We show that in a complex $d$-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have many applications in quantum information theory (e.g.\ in state tomography, cloning, or cryptography) -- however it is suspected that maximal sets exist only in prime-power dimensions. Our construction offers an efficient alternative for general dimensions. The findings are based on a framework recently established, which reduces the construction of such bases to the combinatorial problem of finding certain highly nonlinear functions between abelian groups.

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