scholarly journals Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three

2020 ◽  
Vol 476 (2235) ◽  
pp. 20190754
Author(s):  
Mengyao Hu ◽  
Lin Chen ◽  
Yize Sun
Keyword(s):  
Open Problem ◽  
Unitary Transformation ◽  
Quantum Physics ◽  
Hadamard Matrix ◽  
Mutually Unbiased Bases ◽  
Unitary Operation ◽  
Local Unitary Transformation ◽  
Zero Entry ◽  
Schmidt Rank ◽  
Complex Hadamard Matrix

Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2  ⊗  V and I 2  ⊗  W . We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.

scholarly journals Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

2018 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Association Scheme ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Association Schemes ◽  
Roots Of Unity ◽  
Galois Rings ◽  
Type Complex ◽  
Complex Hadamard Matrix

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

Criteria for a Hadamard Matrix to be Skew-Equivalent

1976 ◽  
Vol 28 (6) ◽  
pp. 1216-1223 ◽  
Author(s):  
Judith Q. Longyear
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Permutation Matrix ◽  
Type I ◽  
Signed Permutation ◽  
Zero Entry

A matrix H of order n = 4t with all entries from the set ﹛1, —1﹜ is Hadamard if HHt = 4tI. The set of Hadamard matrices is . A matrix is of type I or is skew-Hadamard if H = S — I where St = —S (some authors also use H = S + I). The set of type I members is . A matrix P is a signed permutation matrix if each row and each column has exactly one non-zero entry, and that entry is from the set ﹛1, —1﹜.

scholarly journals Complex Hadamard Matrix-Aided Generalized Space Shift Keying Modulation

IEEE Access ◽  
2017 ◽  
Vol 5 ◽  
pp. 21139-21147 ◽  
Author(s):  
Han Hai ◽  
Xue-Qin Jiang ◽  
Wei Duan ◽  
Jun Li ◽  
Moon Ho Lee ◽  
...  
Keyword(s):  
Hadamard Matrix ◽  
Shift Keying ◽  
Complex Hadamard Matrix

scholarly journals Multiplicative functions arising from the study of mutually unbiased bases

10.53733/99 ◽  
2021 ◽  
Vol 51 ◽  
pp. 65-78
Author(s):  
Berthold-Georg Englert ◽  
Heng Huat Chan
Keyword(s):  
Hilbert Space ◽  
Multiplicative Function ◽  
Quantum Physics ◽  
Exponential Sum ◽  
Mutually Unbiased Bases ◽  
Multiplicative Functions

We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.

Mutually unbiased bases and orthogonal decompositions of Lie algebras

2007 ◽  
Vol 7 (4) ◽  
pp. 371-382
Author(s):  
P.O. Boykin ◽  
M. Sitharam ◽  
P.H. Tiep ◽  
P. Wocjan
Keyword(s):  
Lie Algebras ◽  
Open Problem ◽  
Orthogonal Decomposition ◽  
Mutually Unbiased Bases ◽  
Killing Form ◽  
Unitary Matrices ◽  
Complete Collection ◽  
Cartan Subalgebras ◽  
Orthogonal Decompositions ◽  
Adjoint Operation

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of $\mu$ MUBs in $\K^n$ gives rise to a collection of $\mu$ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also $\dag$-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for $n\le 5$ an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for $n=6$ there are at most three monomial MUBs.

On C-Matrices of Arbitrary Powers

1971 ◽  
Vol 23 (3) ◽  
pp. 531-535 ◽  
Author(s):  
Richard J. Turyn
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Main Diagonal ◽  
Quadratic Character ◽  
Symmetric Complex ◽  
Complex Hadamard Matrix ◽  
Square Matrix

A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m > 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)

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