scholarly journals All mutually unbiased bases in dimensions two to five

2010 ◽  
Vol 10 (9&10) ◽  
pp. 803-820
Author(s):  
Stephen Brierley ◽  
Stefan Weigert ◽  
Ingemar Bengtsson
Keyword(s):  
Hadamard Matrices ◽  
Parameter Family ◽  
Mutually Unbiased Bases ◽  
Low Dimensions ◽  
Complete Sets

All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.

Nonisomorphic complete sets of orthogonal f-squares and hadamard matrices

1984 ◽  
Vol 13 (11) ◽  
pp. 1391-1406 ◽  
Author(s):  
S. J. Schwager ◽  
W. T. Federer ◽  
B. L. Raktoe
Keyword(s):  
Hadamard Matrices ◽  
Complete Sets

scholarly journals Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six

2019 ◽  
pp. 1-18 ◽  
Author(s):  
Mengfan Liang ◽  
Mengyao Hu ◽  
Yize Sun ◽  
Lin Chen ◽  
Xiaoyu Chen
Keyword(s):  
Hadamard Matrices ◽  
Mutually Unbiased Bases

scholarly journals Unextendible Sets of Mutually Unbiased Basis Obtained from Complete Subgraphs

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1388
Author(s):  
Andrés García ◽  
Pablo Carlos López
Keyword(s):  
Regular Graph ◽  
Mutually Unbiased Bases ◽  
Complete Sets ◽  
Unbiased Basis

We propose a method, based on the search and identification of complete subgraphs of a regular graph, to obtain sets of Pauli operators whose eigenstates form unextendible complete sets of mutually unbiased bases of n-qubit systems. With this method we can obtain results for complete and inextensible sets of mubs for 2, 3, 4 and 5 qubits.

scholarly journals Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes

10.37236/4915 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Hadi Kharaghani ◽  
Sara Sasani ◽  
Sho Suda
Keyword(s):  
Upper Bound ◽  
Association Scheme ◽  
Hadamard Matrices ◽  
Association Schemes ◽  
Mutually Unbiased Bases ◽  
Identity Matrix ◽  
Polynomial Association Scheme

It was shown by LeCompte, Martin, and Owens in 2010 that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a $Q$-polynomial association scheme of class four which is both $Q$-antipodal and $Q$-bipartite.  We prove that the existence of a set of mutually unbiased Bush-type Hadamard matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain an upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be $2n-1$. This is in contrast to the fact that the best general upper bound for the mutually unbiased Hadamard matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class $4$.

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 ON MUTUALLY UNBIASED BASES

2010 ◽  
Vol 08 (04) ◽  
pp. 535-640 ◽  
Author(s):  
THOMAS DURT ◽  
BERTHOLD-GEORG ENGLERT ◽  
INGEMAR BENGTSSON ◽  
KAROL ŻYCZKOWSKI
Keyword(s):  
Degrees Of Freedom ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Galois Field ◽  
Entanglement Swapping ◽  
Mutually Unbiased Bases ◽  
Existence Problem ◽  
Affine Planes ◽  
Dual Nature ◽  
Open Questions

Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = pM, with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…).In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases.There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime.The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.

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