A complex Hadamard matrix is any complex $N\times N$ matrix $H$ which is unimodular $|H_{jk}|=1$ and unitary $HH^{\dagger}=N\mathbb{1}_N$ up to a constant factor.
To save space we do not describe construction of Hadamard matrices but present a short characterization of each case.
If $H$ is a dephased Hadamard matrix, so are: the transposed matrix $H^{\rm T}$, the conjugated matrix $\bar{H}$ and the Hermitian transpose $H^{\dagger}$.
Starting from a given Hadamard matrix $H$ in the dephased form one may investigate, whether it is possible to perform infinitesimal changes of some of $f=(N-1)^2$ phases of the core of $H$ to preserve unitarity. Assuming that all these phases vary linearly with free parameters one can find analytical form of such orbits, i.e. affine Hadamard families, stemming from e.g. Fourier matrices of composite dimensions. See [38].
If $A$ and $B$ are complex Hadamard matrices of dimension $N$ then $$H=\left[\begin{array}{rr}A&B\\A&-B\end{array}\right]\in\mathcal{H}_{2N}.$$ Furthermore, if $A$ and $B$ are taken to be in the dephased form, so is $H$. This method, originally due to Hadamard, can be generalized by realizing that $B$ can be multiplied at left by an arbitrary diagonal unitary matrix $$E={\rm diag}\Big(1,e^{id_1},\dots,e^{id_{N-1}}\Big).$$ If $A$ and $B$ depend on $a$ and $b$ free parameters, respectively, then $$H'=\left[\begin{array}{rr}A&EB\\A&-EB\end{array}\right]$$ represents an $(a + b + N - 1)$ - parameter family of Hadamard matrices of size $2N$ in the dephased form.
In analogy to previous method one may quadruple the matrix size in a construction similar to that derived by Williamson [93] from quaternions. If $A$, $B$, $C$ and $D$ are complex Hadamard matrices of dimension $N$ in the dephased form then $$\left[\begin{array}{rrrr} A& B& C& D\\ A& -B& C& -D\\ A& B& -C& -D\\ A& -B& -C& D \end{array}\right]\in\mathcal{H}_{4N}$$ and it has the dephased form too. This form is preserved, if the blocks $B$, $C$ and $D$ are multiplied by diagonal unitary matrices $E_1$, $E_2$ and $E_3$ respectively, each containing unity and $f=N - 1$ free phases. Therefore we have an $\left(a + b + c + d + 3(N - 1)\right)$–dimensional family of Hadamard matrices where $a$, $b$, $c$ and $d$ denote the number of free parameters contained in $A$, $B$, $C$ and $D$ respectively.
For any two Hadamard matrices, $A\in\mathcal{H}_K$ and $B\in\mathcal{H}_M$, their tensor product $A\otimes B\in\mathcal{H}_{KM}$. A more general construction by Diţă [56] allows to use apart of a given Hadamard matrix $A$ of size $K$ an entire set of $K$ (possibly different) Hadamard matrices $\{B_1, \dots B_K\}$ of size $M$. Then the matrix $$\left[\begin{array}{cccc} A_{11}B_1&A_{12}E_2B_2&\cdots&A_{1K}E_KB_K\\ A_{21}B_1&A_{22}E_2B_2&\cdots&A_{2K}E_KB_K\\ \vdots& \vdots& &\vdots\\ A_{K1}B_1&A_{K2}E_2B_2&\cdots&A_{KK}E_KB_K \end{array}\right]$$ of size $N=KM$ is Hadamard. We may introduce additional free phases by using $m=K - 1$ diagonal unitary matrices $E_k$. The total number of free phases reads $f=(M - 1)(K - 1) + a + b_1 + ... + b_K$ where $a$ and $b_i$ denote the numbers of free phases used in the definitions of $A_i$ and $B_i$ respectively.
Details are described in the paper of Petre Diţă [103].
Details of the construction are included in the work by M. Matolcsi, J. Réffy and F. Szöllősi, see [34].
Consider $\{A_1, A_2, ..., A_n\}$ a set of CHM, each of order $m\geqslant 2$, and $\{B_1, B_2, ...,B_m\}$ - each of order $n\geqslant 2$. Matrix $M$ of order $mn\times mn$ is defined as $M=[u_{jk} v_{jk}]$, where $u_{jk}$ is $k^{\rm th}$ column of $A_j$ and $v_{jk}$ is $j^{\rm th}$ row of $B_k$. Matrix $M$ consists of $m\times n$ rank-one blocks generated by "weaving" appropriate rows and columns from $A_j$ and $B_k$.
Suppose each $A_j$ has affine dimension $a_j$ and each $B_k$ has affine dimension $b_k$ (both after dephasing). Then, matrix $M$ generated by weaving $A_j$ and $B_k$ yields (again, after dephasing) $$f=\sum_j a_j + \sum_k b_k +(m-1)(n-1)$$ free parameters.
Matlab script to generate weaving Hadamard matrices is available on GitHub.
Details are provided in the Master Thesis by Marteen Havinga [176].
Details are provided in the paper of Santiago Barrera Acevedo, Heiko Dietrich and Corey Lionis [179].
Complex Hadamard matrices play a crucial role in the theory of quantum information as it is shown in a seminal paper of Werner [92]. They are used in solving the Mean King Problem [16], [59] to construct "nice error basis" [75], [17] or "quantum designs" [65], [97]. Furthermore, they allow one to construct:
These problems are equivalent in the sense that given a solution to one problem one can find a solution to the other one, as well as a corresponding scheme of teleportation or dense coding.
Another application of Hadamard matrices is related to quantum tomography: to determine all $m=N^2 - 1$ parameters characterizing a density matrix of size $N$ one needs to perform $k > N$ orthogonal measurements. Each measurement can be specified by an orthogonal basis $\Phi_u=\{|\varphi_i^{(u)}\rangle\}$ set for $u=1,2,...,k$. Precision of such a measurement scheme is optimal if the bases are mutually unbiased, i.e. they are such that $$|\langle\varphi_i^{(u)}|\varphi_j^{(s)}\rangle|^2=\delta_{us}\delta_{ij}+(1-\delta_{us})/N.$$ The task of finding $(k + 1)$ MUBs is equivalent to finding a collection of $k$ mutually unbiased Hadamards (MUHs) $$\{H_i\in\mathcal{H}_N\}_{i=1,2,...,k}:\frac{1}{\sqrt{N}}H_i^{\dagger}H_j\in\mathcal{H}_N \quad(i>j=1,2,...,k-1),$$ since the set $\{\mathbb{1}_N,\frac{1}{\sqrt{N}}H_1,...,\frac{1}{\sqrt{N}}H_N\}$ forms a set of MUBs.
If $N$ is a prime or a power of prime there exist a complete set of $m=N + 1$ MUBs which provide an optimal scheme of quantum tomography [68], [95]. If $N$ is not a power of prime the problem of specifying the maximal number of MUBs remains open [12], [75], [29], [7], [99], [40], [43].
Note that the presented list of equivalence classes is complete only for $N =$ $2$, $3$, $4$ and $5$ while for $N > 5$ the full set of solutions remains unknown. The list of open questions could be rather long, but let us mention here some most relevant:
→ Szöllősi showed a general construction [25] that for $N\geqslant$ $12$, real Hadamard matrices belong to continuous families.
→ A partial answer was given in May 2006 by Beauchamp and Nicoară [3], who found a non affine family for $N=6$.
Problems analogous to 1, 2, 3 and 4 are obviously open for higher dimensions. Thus a lot of work is still required to get a full understanding of the properties of the set of complex Hadamard matrices, even for one-digit dimensions.
Interestingly, the dimension $N=6$, the smallest product of two different primes, is the first case for which not all complex Hadamard matrices are known, as well as the simplest case for which the MUB problem remains open [12], [29], [99], [40], [43].