--- title: Hadamard-Phase Parametrization url: https://www.emergentmind.com/topics/hadamard-phase-parametrization type: topic --- # Hadamard-Phase Parametrization Hadamard-phase parametrization is a method for describing families of unitary matrices—and therefore orthonormal bases—by attaching phase variables to a fixed complex Hadamard matrix. In its basic form one writes \[ U(\theta)=\frac{1}{\sqrt d}\,H\,\operatorname{diag}(e^{i\theta_1},\dots,e^{i\theta_d}), \] often after fixing one phase by a gauge convention. Because a complex Hadamard matrix has constant-modulus entries and orthogonal columns, every such \(U(\theta)\) is unitary and its columns form a basis mutually unbiased with respect to the computational basis. In the modern MUB literature, this parametrization turns mutual-unbiasedness questions into explicit constraints on phase differences; in the broader theory of complex Hadamard matrices, it also appears as a general phase-coordinate system for affine and non-affine Hadamard families [2604.02234]. ## 1. Definition and algebraic setting In \(\mathbb{C}^d\), a complex Hadamard matrix \(H\) is a \(d\times d\) matrix such that every entry has modulus \(1\) and \(\frac{1}{\sqrt d}H\) is unitary. If \(U=\frac{1}{\sqrt d}H\) and \(\{|f_k\rangle\}\) denotes the columns of \(U\), then every coefficient of \(|f_k\rangle\) in the computational basis \(\{|e_j\rangle\}\) has modulus \(1/\sqrt d\), so \[ |\langle e_j\mid f_k\rangle|^2=\frac{1}{d}. \] Therefore the computational basis and the Hadamard basis are mutually unbiased. Sets of mutually unbiased bases (MUBs) can then be described by sets of such unitaries whose pairwise transition matrices are themselves complex Hadamard matrices [2604.02234]. Hadamard-phase parametrization fixes a reference Hadamard matrix and varies only phase degrees of freedom. In the formulation emphasized for MUBs, the variation is by right multiplication with a diagonal phase matrix, so orthogonality of columns and constant entry modulus are preserved automatically. In the broader complex Hadamard literature, the same idea is often written entrywise: if a dephased Hadamard matrix \(H\) is fixed, one considers families of the form \(H\circ \mathrm{EXP}(iR)\), where \(R\) is a real phase matrix with zero first row and column. This places the nontrivial continuous parameters in the core of the matrix and separates genuine deformations from row and column gauge phases [1110.5590]. ## 2. Dimension \(4\): the canonical diagonal-phase family The most explicit finite-dimensional realization is the \(d=4\) construction based on \[ H_4=H_2\otimes H_2= \begin{pmatrix} 1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1 \end{pmatrix}, \qquad H_2= \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}. \] Since \(\tfrac12 H_4\) is unitary, its columns define one unbiased basis. A family of bases is obtained by multiplying on the right by \[ D(\alpha,\beta,\gamma)=\operatorname{diag}(1,e^{i\alpha},e^{i\beta},e^{i\gamma}), \] and setting \[ \mathcal{B}(\theta)=\frac12 H_4 D(\theta),\qquad \theta=(\alpha,\beta,\gamma)\in\mathbb{T}^3. \] The first phase is fixed to \(1\) because a global column phase does not change the basis, so the family lives on a \(3\)-torus rather than \(\mathbb{T}^4\). For every \(\theta\), \(\mathcal{B}(\theta)\) is an orthonormal basis and every coefficient in the computational basis has modulus \(1/2\), hence every \(\mathcal{B}(\theta)\) is automatically mutually unbiased with the computational basis [2604.02234]. The nontrivial question is when two different phase choices \(\theta\) and \(\theta'\) produce bases that are mutually unbiased with each other. Writing \(\Delta_\alpha=\alpha'-\alpha\), \(\Delta_\beta=\beta'-\beta\), and \(\Delta_\gamma=\gamma'-\gamma\), the overlaps take the form \[ \langle v_i(\theta)\mid v_j(\theta')\rangle = \frac14\Bigl(1+\varepsilon_2 e^{i\Delta_\alpha}+\varepsilon_3 e^{i\Delta_\beta}+\varepsilon_4 e^{i\Delta_\gamma}\Bigr), \] where \(\varepsilon_k\in\{\pm1\}\) depends on the column pair \((i,j)\). Mutual unbiasedness is therefore equivalent to a trigonometric constraint on phase differences: \[ \varepsilon_2 \cos\Delta_\alpha + \varepsilon_3 \cos\Delta_\beta + \varepsilon_4 \cos\Delta_\gamma + \varepsilon_2\varepsilon_3 \cos(\Delta_\alpha-\Delta_\beta) + \varepsilon_2\varepsilon_4 \cos(\Delta_\alpha-\Delta_\gamma) + \varepsilon_3\varepsilon_4 \cos(\Delta_\beta-\Delta_\gamma)=0. \] In the symmetric case \(\Delta_\alpha=\Delta_\beta=\Delta_\gamma=\Delta\), the overlap reduces to \(\frac14(1+k e^{i\Delta})\) with \(k\in\{-3,-1,1,3\}\), and explicit solutions such as \(\Delta=\pm \pi/2\) occur for certain sign patterns. The paper stresses that this reduced parametrization exhibits continuous phase orbits: continuous sets of \((\alpha,\beta,\gamma)\) remain unbiased to the computational basis, and certain continuous sets produce mutually unbiased pairs inside the family [2604.02234]. ## 3. From phase families to complete MUB sets in \(d=4\) The diagonal-phase family does more than generate isolated unbiased bases. In \(d=4\), special phase choices recover a complete set of \(5\) MUBs. Besides the computational basis \(\mathcal{B}_0\), the construction uses \[ D_2=\operatorname{diag}(1,i,-1,i),\quad D_3=\operatorname{diag}(1,-1,1,-1),\quad D_4=\operatorname{diag}(1,-i,-1,i), \] and defines \[ \mathcal{B}_k=\frac12 H_4 D_k,\qquad k=1,2,3,4. \] Direct overlap calculations verify that \[ \{\mathcal{B}_0,\mathcal{B}_1,\mathcal{B}_2,\mathcal{B}_3,\mathcal{B}_4\} \] is a complete set of \(5\) MUBs in dimension \(4\) [2604.02234]. The same structure can be expressed in Pauli-group language. Using the identification \(\mathbb{C}^4\simeq \mathbb{C}^2\otimes\mathbb{C}^2\), one considers commuting families of two-qubit Pauli operators such as \(\{\sigma_z\otimes I,\,I\otimes \sigma_z\}\) and \(\{\sigma_x\otimes I,\,I\otimes \sigma_x\}\). Their common eigenbases form a complete set of \(5\) MUBs. The relation to the phase picture is explicit: \(H_2\) diagonalizes \(\sigma_x\), so \(H_4=H_2\otimes H_2\) diagonalizes \(\sigma_x\otimes I\) and \(I\otimes \sigma_x\), while the diagonal matrices \(D_k\) add phases in the corresponding eigenbasis. In this sense, the Hadamard-phase parametrization serves as a concrete coordinate system for Pauli-derived MUBs in \(d=4\) [2604.02234]. A common misunderstanding is to treat the continuous \(3\)-torus of column phases as though it already encoded arbitrary \(4\)-dimensional MUB structure. The \(d=4\) result is more specific: it gives a tractable phase chart around a fixed Hadamard scaffold, and the completeness of the \(5\)-MUB set ultimately reflects the tensor-product and Pauli algebra of two qubits rather than diagonal phases alone. ## 4. Dimension \(6\): Fourier-family formulation and rigidity The same philosophy extends formally to \(d=6\), but with markedly different behavior. The construction begins from the \(6\times6\) Fourier matrix \[ [F_6]_{jk}=\frac{1}{\sqrt6}\,\omega^{jk},\qquad \omega=e^{2\pi i/6}, \] and sets \[ \mathcal{B}_\theta=F_6\,\operatorname{diag}(1,e^{i\theta_1},\dots,e^{i\theta_5}),\qquad \theta\in\mathbb{T}^5. \] As in \(d=4\), every \(\mathcal{B}_\theta\) is automatically unbiased with respect to the computational basis because all entries have modulus \(1/\sqrt6\) [2604.02234]. For two phase vectors \(\theta,\theta'\), however, mutual unbiasedness requires \[ \left|\frac16\sum_{k=0}^{5} e^{i(\theta'_k-\theta_k)}\omega^{k(j-i)}\right|^2=\frac16 \qquad \forall i,j, \] with \(\theta_0\equiv0\). This is a system of \(36\) nonlinear equations on phase differences. Formally the structure resembles the \(d=4\) case—overlaps are weighted sums of unit-modulus phase factors—but structurally the constraints are far more rigid. The paper attributes this to the absence of a prime-power structure and the lack of a simple tensor-product decomposition analogous to \(H_2\otimes H_2\) [2604.02234]. The practical consequence is well known in the MUB problem and is sharpened by the phase viewpoint: only up to \(3\) MUBs are known in dimension \(6\), and no choice of \(\theta\) is known that extends a \(3\)-MUB set to \(4\) or more while satisfying all constraints. The same rigidity is reflected in the existence of isolated complex Hadamard matrices in dimension \(6\), such as the Tao matrix. The Hadamard-phase parametrization therefore clarifies a central limitation: it is an effective diagnostic of rigidity even where it does not yield a complete construction [2604.02234]. ## 5. Generalizations in prime-power dimensions and complex Hadamard theory For \(d=p^n\), the MUB problem has a different algebraic character. In prime dimension one defines Heisenberg–Weyl operators \[ X|k\rangle=|k+1\bmod d\rangle,\qquad Z|k\rangle=\omega^k|k\rangle,\qquad \omega=e^{2\pi i/d}, \] which satisfy \(ZX=\omega XZ\). The eigenbases of \(Z, X, XZ, XZ^2,\dots,XZ^{d-1}\) form a complete set of \(d+1\) MUBs. In finite-field form for \(d=p^n\), basis vectors can be written as \[ |v_b^{(a)}\rangle=\frac{1}{\sqrt d}\sum_{x\in\mathbb{F}_{p^n}} \omega^{\operatorname{Tr}(ax^2+bx)}|x\rangle. \] These bases again have constant modulus \(1/\sqrt d\), but their phases are quadratic functions on the finite field rather than independent diagonal column phases. The \(d=4\) diagonal-phase picture thus generalizes to finite-field-structured phase functions in arbitrary prime-power dimensions [2604.02234]. The broader theory of complex Hadamard matrices recasts this idea as a phase-geometry on dephased matrices. Infinitesimal phase deformations of a dephased Hadamard \(H\) are governed by the linearized system \[ \sum_{k=1}^{n} H_{i,k}\overline{H}_{j,k}\,(R_{i,k}-R_{j,k})=0,\qquad 1\le i