About mutually unbiased bases in even and odd prime power dimensions

2005 ◽  
Vol 38 (23) ◽  
pp. 5267-5283 ◽  
Author(s):  
Thomas Durt
Keyword(s):  
Prime Power ◽  
Mutually Unbiased Bases

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.

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.

scholarly journals Galois unitaries, mutually unbiased bases, and mub-balanced states

2015 ◽  
Vol 15 (15&16) ◽  
pp. 1261-1294
Author(s):  
D. Marcus Appleby ◽  
Ingermar Bengtsson ◽  
Hoan Bui Dang
Keyword(s):  
Hilbert Space ◽  
Number Field ◽  
Prime Power ◽  
Cyclotomic Field ◽  
Projective Group ◽  
Mutually Unbiased Bases ◽  
Unitary Operators ◽  
Rotationally Symmetric ◽  
The One ◽  
Symmetric States

A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a cyclotomic field. By including Galois unitaries we are able to remove a mismatch between the finite projective group acting on the bases on the one hand, and the set of those permutations of the bases that can be implemented as transformations in Hilbert space on the other hand. In particular we show that there exist transformations that cycle through all the bases in all dimensions d = p^n where p is an odd prime and the exponent n is odd. (For even primes unitary MUB-cyclers exist.) These transformations have eigenvectors, which are MUB-balanced states (i.e. rotationally symmetric states in the original terminology of Wootters and Sussman) if and only if d = 3 modulo 4. We conjecture that this construction yields all such states in odd prime power dimension.

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