Hadamard-Phase Parametrization

• Hadamard-phase parametrization is a method to attach phase variables to a fixed complex Hadamard matrix, yielding unitary matrices whose columns are mutually unbiased.
• It converts phase variations into explicit trigonometric constraints that simplify the analysis and construction of mutually unbiased bases, particularly in prime-power dimensions.
• The method highlights structural insights, enabling complete MUB sets in dimension 4 while revealing rigidity in non-prime-power dimensions like 6.

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 (Pain, 2 Apr 2026).

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 $U(\theta)$0 in the computational basis $U(\theta)$1 has modulus $U(\theta)$2, so

$U(\theta)$3

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 (Pain, 2 Apr 2026).

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 $U(\theta)$4 is fixed, one considers families of the form $U(\theta)$5, where $U(\theta)$6 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 (Szöllősi, 2011).

2. Dimension $U(\theta)$7: the canonical diagonal-phase family

The most explicit finite-dimensional realization is the $U(\theta)$8 construction based on

$U(\theta)$9

Since $\mathbb{C}^d$0 is unitary, its columns define one unbiased basis. A family of bases is obtained by multiplying on the right by

$\mathbb{C}^d$1

and setting

$\mathbb{C}^d$2

The first phase is fixed to $\mathbb{C}^d$3 because a global column phase does not change the basis, so the family lives on a $\mathbb{C}^d$4-torus rather than $\mathbb{C}^d$5. For every $\mathbb{C}^d$6, $\mathbb{C}^d$7 is an orthonormal basis and every coefficient in the computational basis has modulus $\mathbb{C}^d$8, hence every $\mathbb{C}^d$9 is automatically mutually unbiased with the computational basis (Pain, 2 Apr 2026).

The nontrivial question is when two different phase choices $H$0 and $H$1 produce bases that are mutually unbiased with each other. Writing $H$2, $H$3, and $H$4, the overlaps take the form

$H$5

where $H$6 depends on the column pair $H$7. Mutual unbiasedness is therefore equivalent to a trigonometric constraint on phase differences: $H$8 In the symmetric case $H$9, the overlap reduces to $d\times d$0 with $d\times d$1, and explicit solutions such as $d\times d$2 occur for certain sign patterns. The paper stresses that this reduced parametrization exhibits continuous phase orbits: continuous sets of $d\times d$3 remain unbiased to the computational basis, and certain continuous sets produce mutually unbiased pairs inside the family (Pain, 2 Apr 2026).

3. From phase families to complete MUB sets in $d\times d$4

The diagonal-phase family does more than generate isolated unbiased bases. In $d\times d$5, special phase choices recover a complete set of $d\times d$6 MUBs. Besides the computational basis $d\times d$7, the construction uses

$d\times d$8

and defines

$d\times d$9

Direct overlap calculations verify that

$1$0

is a complete set of $1$1 MUBs in dimension $1$2 (Pain, 2 Apr 2026).

The same structure can be expressed in Pauli-group language. Using the identification $1$3, one considers commuting families of two-qubit Pauli operators such as $1$4 and $1$5. Their common eigenbases form a complete set of $1$6 MUBs. The relation to the phase picture is explicit: $1$7 diagonalizes $1$8, so $1$9 diagonalizes $\frac{1}{\sqrt d}H$0 and $\frac{1}{\sqrt d}H$1, while the diagonal matrices $\frac{1}{\sqrt d}H$2 add phases in the corresponding eigenbasis. In this sense, the Hadamard-phase parametrization serves as a concrete coordinate system for Pauli-derived MUBs in $\frac{1}{\sqrt d}H$3 (Pain, 2 Apr 2026).

A common misunderstanding is to treat the continuous $\frac{1}{\sqrt d}H$4-torus of column phases as though it already encoded arbitrary $\frac{1}{\sqrt d}H$5-dimensional MUB structure. The $\frac{1}{\sqrt d}H$6 result is more specific: it gives a tractable phase chart around a fixed Hadamard scaffold, and the completeness of the $\frac{1}{\sqrt d}H$7-MUB set ultimately reflects the tensor-product and Pauli algebra of two qubits rather than diagonal phases alone.

4. Dimension $\frac{1}{\sqrt d}H$8: Fourier-family formulation and rigidity

The same philosophy extends formally to $\frac{1}{\sqrt d}H$9, but with markedly different behavior. The construction begins from the $U=\frac{1}{\sqrt d}H$0 Fourier matrix

$U=\frac{1}{\sqrt d}H$1

and sets

$U=\frac{1}{\sqrt d}H$2

As in $U=\frac{1}{\sqrt d}H$3, every $U=\frac{1}{\sqrt d}H$4 is automatically unbiased with respect to the computational basis because all entries have modulus $U=\frac{1}{\sqrt d}H$5 (Pain, 2 Apr 2026).

For two phase vectors $U=\frac{1}{\sqrt d}H$6, however, mutual unbiasedness requires

$U=\frac{1}{\sqrt d}H$7

with $U=\frac{1}{\sqrt d}H$8. This is a system of $U=\frac{1}{\sqrt d}H$9 nonlinear equations on phase differences. Formally the structure resembles the $\{|f_k\rangle\}$0 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 $\{|f_k\rangle\}$1 (Pain, 2 Apr 2026).

The practical consequence is well known in the MUB problem and is sharpened by the phase viewpoint: only up to $\{|f_k\rangle\}$2 MUBs are known in dimension $\{|f_k\rangle\}$3, and no choice of $\{|f_k\rangle\}$4 is known that extends a $\{|f_k\rangle\}$5-MUB set to $\{|f_k\rangle\}$6 or more while satisfying all constraints. The same rigidity is reflected in the existence of isolated complex Hadamard matrices in dimension $\{|f_k\rangle\}$7, 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 (Pain, 2 Apr 2026).

5. Generalizations in prime-power dimensions and complex Hadamard theory

For $\{|f_k\rangle\}$8, the MUB problem has a different algebraic character. In prime dimension one defines Heisenberg–Weyl operators

$\{|f_k\rangle\}$9

which satisfy $U$0. The eigenbases of $U$1 form a complete set of $U$2 MUBs. In finite-field form for $U$3, basis vectors can be written as

$U$4

These bases again have constant modulus $U$5, but their phases are quadratic functions on the finite field rather than independent diagonal column phases. The $U$6 diagonal-phase picture thus generalizes to finite-field-structured phase functions in arbitrary prime-power dimensions (Pain, 2 Apr 2026).

The broader theory of complex Hadamard matrices recasts this idea as a phase-geometry on dephased matrices. Infinitesimal phase deformations of a dephased Hadamard $U$7 are governed by the linearized system

$U$8

and the associated defect gives an upper bound on the local dimension of smooth Hadamard families through $U$9. Concrete parametrization schemes include row-pair phase insertions, block-structured phase deformations, and affine families of the form $U(\theta)$00 (Szöllősi, 2011, Lampio et al., 2012).

A particularly explicit higher-order example is the order-$U(\theta)$01 two-unitary complex Hadamard family

$U(\theta)$02

where $U(\theta)$03 is a fixed $U(\theta)$04 integer matrix with entries in $U(\theta)$05 and $U(\theta)$06 is an integer-valued phase matrix depending affinely on $U(\theta)$07 parameters. This realizes a $U(\theta)$08-dimensional affine family of two-unitary complex Hadamard matrices inside a sixth-root-of-unity phase structure (Bruzda et al., 2024).

Context	Parametrized object	Phase structure
$U(\theta)$09 MUBs	$U(\theta)$10	3 independent column phases
$U(\theta)$11 MUBs	$U(\theta)$12	5 independent column phases
Order-$U(\theta)$13 CHM	$U(\theta)$14	sixth-root seed plus 19 affine parameters

6. Terminological scope, applications, and limitations

The term is not fully uniform across fields. In complex Hadamard and MUB theory, Hadamard-phase parametrization refers to phase variables attached to a fixed Hadamard scaffold, typically by diagonal column phases or entrywise phase matrices. In optical communication, a related construction appears in Hadamard-word binary phase-shift keying, where codeword amplitudes are written as

$U(\theta)$15

so that the receiver analyzes interference sums of the form $U(\theta)$16. There the phase parameters model physical fluctuations rather than MUB coordinates, but the Hadamard structure again turns phase perturbations into explicit interference constraints (Jarzyna et al., 2015).

In optimization, the terminology shifts further. “Hadamard parametrization” may denote over-parameterization by elementwise products or differences of squares, such as $U(\theta)$17, $U(\theta)$18, or $U(\theta)$19, in sparse phase retrieval, $U(\theta)$20-regularized models, and polyhedral optimization. These constructions preserve the Hadamard, i.e. entrywise, architecture, but they do not parametrize unit-modulus phase variables in the sense used for complex Hadamard matrices and MUBs (Wu et al., 2020, Ouyang et al., 2024, Tang et al., 2024). This suggests a terminological continuity centered on entrywise structure rather than a single universal definition.

The principal limitation of Hadamard-phase parametrization is therefore structural, not formal. It is highly effective when the underlying dimension carries tensor-product, Pauli, or finite-field symmetries, because those symmetries translate into solvable phase equations. It becomes rigid in dimensions such as $U(\theta)$21, where the same equations appear over-determined and no complete MUB construction is known. For that reason the parametrization is best understood not merely as a recipe for inserting phases, but as a diagnostic language for how algebraic structure, gauge freedom, and geometric interference interact in the existence—or obstruction—of complex Hadamard and MUB families (Pain, 2 Apr 2026).