Complex Hadamard matrix
A complex Hadamard matrix is any complex [math]\displaystyle{ N \times N }[/math] matrix [math]\displaystyle{ H }[/math] satisfying two conditions:
- unimodularity (the modulus of each entry is unity): [math]\displaystyle{ |H_{jk}| = 1 \text{ for } j,k = 1,2,\dots,N }[/math]
- orthogonality: [math]\displaystyle{ HH^{\dagger} = NI }[/math],
where [math]\displaystyle{ \dagger }[/math] denotes the Hermitian transpose of [math]\displaystyle{ H }[/math] and [math]\displaystyle{ I }[/math] is the identity matrix. The concept is a generalization of Hadamard matrices. Note that any complex Hadamard matrix [math]\displaystyle{ H }[/math] can be made into a unitary matrix by multiplying it by [math]\displaystyle{ \frac{1}{\sqrt{N}} }[/math]; conversely, any unitary matrix whose entries all have modulus [math]\displaystyle{ \frac{1}{\sqrt{N}} }[/math] becomes a complex Hadamard upon multiplication by [math]\displaystyle{ \sqrt{N}. }[/math]
Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation. Real Hadamard matrices and Butson-type Hadamard matrices form particular cases of complex Hadamard matrices.
Complex Hadamard matrices exist for any natural number [math]\displaystyle{ N }[/math] (compare with the real case, in which Hadamard matrices do not exist for every [math]\displaystyle{ N }[/math] and existence is not known for every permissible [math]\displaystyle{ N }[/math]). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor),
- [math]\displaystyle{ [F_N]_{jk}:= \exp[2\pi i (j-1)(k-1)/N] {\quad \rm for \quad} j,k=1,2,\dots,N }[/math]
belong to this class.
Equivalency
Two complex Hadamard matrices are called equivalent, written [math]\displaystyle{ H_1 \simeq H_2 }[/math], if there exist diagonal unitary matrices [math]\displaystyle{ D_1, D_2 }[/math] and permutation matrices [math]\displaystyle{ P_1, P_2 }[/math] such that
- [math]\displaystyle{ H_1 = D_1 P_1 H_2 P_2 D_2. }[/math]
Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.
For [math]\displaystyle{ N=2,3 }[/math] and [math]\displaystyle{ 5 }[/math] all complex Hadamard matrices are equivalent to the Fourier matrix [math]\displaystyle{ F_{N} }[/math]. For [math]\displaystyle{ N=4 }[/math] there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,
- [math]\displaystyle{ F_{4}^{(1)}(a):= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & ie^{ia} & -1 & -ie^{ia} \\ 1 & -1 & 1 &-1 \\ 1 & -ie^{ia}& -1 & i e^{ia} \end{bmatrix} {\quad \rm with \quad } a\in [0,\pi) . }[/math]
For [math]\displaystyle{ N=6 }[/math] the following families of complex Hadamard matrices are known:
- a single two-parameter family which includes [math]\displaystyle{ F_6 }[/math],
- a single one-parameter family [math]\displaystyle{ D_6(t) }[/math],
- a one-parameter orbit [math]\displaystyle{ B_6(\theta) }[/math], including the circulant Hadamard matrix [math]\displaystyle{ C_6 }[/math],
- a two-parameter orbit including the previous two examples [math]\displaystyle{ X_6(\alpha) }[/math],
- a one-parameter orbit [math]\displaystyle{ M_6(x) }[/math] of symmetric matrices,
- a two-parameter orbit including the previous example [math]\displaystyle{ K_6(x,y) }[/math],
- a three-parameter orbit including all the previous examples [math]\displaystyle{ K_6(x,y,z) }[/math],
- a further construction with four degrees of freedom, [math]\displaystyle{ G_6 }[/math], yielding other examples than [math]\displaystyle{ K_6(x,y,z) }[/math],
- a single point - one of the Butson-type Hadamard matrices, [math]\displaystyle{ S_6 \in H(3,6) }[/math].
It is not known, however, if this list is complete, but it is conjectured that [math]\displaystyle{ K_6(x,y,z),G_6,S_6 }[/math] is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.
References
- U. Haagerup, Orthogonal maximal abelian *-subalgebras of the n×n matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (Rome), 1996 (Cambridge, MA: International Press) pp 296–322.
- P. Dita, Some results on the parametrization of complex Hadamard matrices, J. Phys. A: Math. Gen. 37, 5355-5374 (2004).
- F. Szollosi, A two-parametric family of complex Hadamard matrices of order 6 induced by hypocycloids, preprint, arXiv:0811.3930v2 [math.OA]
- W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices Open Systems & Infor. Dyn. 13 133-177 (2006)
External links
- For an explicit list of known [math]\displaystyle{ N=6 }[/math] complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices
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