Complex Basis Transformations

• Complex basis transformations are invertible linear changes of basis using complex-valued matrices that preserve algebraic structures while enabling improved diagonalization and symmetry exploitation.
• They are applied across quantum mechanics, machine learning, signal processing, and distributed control to optimize computations and maintain invariance under transformation.
• Techniques such as matrix exponentials, unitary rotations, and Householder-type reflections facilitate robust and efficient numerical methods and theoretical models.

Complex basis transformations refer to invertible linear changes of basis generated by complex-valued matrices or operators, most commonly unitary or orthogonal, which act on vector spaces, Hilbert spaces, operator algebras, or data representations. These transformations preserve the vector space or algebraic structure while enabling a change of perspective—typically to improve diagonalization, exploit symmetry, enable invariance, or facilitate computation. At the core, complex basis transformations are ubiquitous across mathematics, physics, quantum theory, signal processing, machine learning, and distributed control.

1. Algebraic Formulation and Matrix Exponential Approach

A fundamental approach to basis transformations in complex linear spaces leverages the matrix exponential form, notably in the context of the discrete Fourier transform (DFT) and its hypercomplex generalizations. Given a real or complex $N \times N$ matrix $J$ satisfying $J^2 = -I_N$, one constructs the exponential

$e^{J\theta} = I_n \cos\theta + J\sin\theta,$

generalizing Euler's formula to arbitrary (possibly hypercomplex) algebras. This machinery enables discrete transforms—including complex, quaternionic, and Clifford DFTs—to be unified via matrix exponentials. The transformation $e^{J\theta}$ acts as a basis rotation or hyper-rotation on the underlying vector or function space, efficiently computable using standard matrix routines without requiring special algebra-specific code (Sangwine et al., 2010).

In this framework, the basis change (and thus the structure of the DFT) is entirely encoded in the choice of matrix root $J$ of $-1$. For example, $J$ corresponding to the $2 \times 2$ real representation of $j$ implements a planar rotation in $J$0, while higher-dimensional $J$1 matrices enable rotations or transformations in $J$2 or higher, as appropriate for quaternionic or Clifford-valued data (Sangwine et al., 2010).

2. Complex Basis Transformations in Quantum and Classical Systems

Complex basis transformations are central to quantum spin, field theory, and mixed quantum-classical dynamics. In the latter, canonical position and momentum variables $J$3 can be combined into complex-valued classical coordinates $J$4, permitting arbitrary unitary transformations $J$5 with $J$6.

When these transformations are applied, all quadratic and higher-order terms in the Hamiltonian (and the quantum-classical coupling operators) are correspondingly mapped. The dynamics of both quantum amplitudes and the classical phase space are then evolved in the new basis:

• Quantum equations use the instantaneous Hamiltonian in the rotated basis,
• Classical equations are derived as Hamilton equations for $J$7.

The main advantage is the ability to localize or delocalize coordinates and to heavily truncate both quantum and classical basis sets while preserving essential features of the dynamics. For example, in surface hopping descriptions of electron-phonon scattering with localized impurities, the transformation to an electronic eigenbasis combined with complexified classical coordinates yields population dynamics nearly identical to the fully untruncated calculation, but with computational cost reduced by an order of magnitude (Miyazaki et al., 2024).

3. Complex Basis Invariance and Transformation Learning in Machine Learning

Complex basis transformations and invariance are exploited in modern representation learning. In the Complex Autoencoder (CAE) architecture, real data vectors $J$8 are projected to a complex space $J$9 by learnable complex weights $J^2 = -I_N$0, yielding $J^2 = -I_N$1. When the weight matrix $J^2 = -I_N$2 is (learned to be) approximately unitary and aligned with the symmetry group of interest (e.g., time-shifts or rotation), the magnitude $J^2 = -I_N$3 of the complex coefficients is strictly invariant under orthogonal transformations of the input.

For an orthogonal transformation $J^2 = -I_N$4, there exists a corresponding unitary $J^2 = -I_N$5 and diagonal $J^2 = -I_N$6 with unimodular entries such that $J^2 = -I_N$7, implying $J^2 = -I_N$8. Thus, the CAE's magnitude channel encodes a transformation-invariant feature space, while the phase encodes the particular transformation applied. This dual encoding supports invariance in tasks like music structure discovery and audio alignment, where phase differences can also be harnessed to infer the specific transformation relating two signals (Lattner et al., 2019).

4. Symmetry, Invariance, and Complex GNNs in Distributed Systems

In distributed multi-agent (e.g., robotic) control and learning, complex basis transformations implement rotation and frame changes naturally via the identification of $J^2 = -I_N$9, with $e^{J\theta} = I_n \cos\theta + J\sin\theta,$0 and rotations corresponding to multiplication by $e^{J\theta} = I_n \cos\theta + J\sin\theta,$1. This enables the construction of Graph Neural Networks (GNNs) that are strictly invariant to arbitrary local agent frame choices.

Each message-passing layer in such a GNN rotates all geometric features and messages into a common local frame, applies SO(2)-equivariant complex linear and activation operations, and guarantees that, when observed from a global frame, the outputs are invariant to all local basis conventions. The composition of phase-equivariant layers preserves this global invariance, dramatically improving the robustness and generalization of distributed policies in challenging environments such as GPS-denied navigation (Honor et al., 3 Apr 2026).

5. Complex Basis Transformations in Quantum Information and Symmetry Analysis

The fundamental role of complex basis transformations pervades quantum information protocols and symmetry analysis in quantum field models. In entanglement-swapping-based quantum key distribution (QKD), optimal basis transformations can be expressed as general SU(2) rotations of the form

$e^{J\theta} = I_n \cos\theta + J\sin\theta,$2

where optimal security is achieved by tuning the angles $e^{J\theta} = I_n \cos\theta + J\sin\theta,$3. For both single- and double-party transformations, one can quantitatively optimize the choice to minimize adversary mutual information and maximize the error rate forced upon an eavesdropper. Crucially, moving beyond the standard Hadamard choice can halve or further reduce leaked information (Schauer et al., 2013).

In the context of the complex two-Higgs doublet model (C2HDM), the physical content must be invariant under arbitrary U(2) complex basis (flavor) rotations of the Higgs fields. The basis-independent formalism constructs quantities (projectors, invariants, pseudo-invariants) that transform covariantly under U(2), ensuring the model’s parameterization and phenomenology remain independent of basis choices. The Higgs basis itself is unique up to a complex phase, and the presence/absence of explicit or spontaneous CP-violation is diagnosed via basis-invariant combinations of potential parameters (Boto et al., 2020).

6. Numerical Algorithms and Householder-Type Transformations

In computational linear algebra, certain classes of matrices (notably complex symmetric) require similarity transformations with complex-orthogonal, rather than unitary, matrices for diagonalization. The generalized Householder reflector, formulated with respect to a non-conjugate (bilinear) inner product,

$e^{J\theta} = I_n \cos\theta + J\sin\theta,$4

operates as a complex-symmetric, orthogonal (but not unitary) transformation. Such reflectors preserve the symmetry of the matrix under $e^{J\theta} = I_n \cos\theta + J\sin\theta,$5, enabling efficient tridiagonalization and subsequent diagonalization of complex symmetric matrices arising in pseudo-Hermitian quantum mechanics and related domains (Noble et al., 2013).

7. Computational and Structural Implications

Adoption of matrix-exponential, unitary, or orthogonal complex basis transformations enables unification of signal-processing methods across associative hypercomplex algebras, reference implementations using only standard matrix routines, and significant algorithmic simplification—obviating the need for bespoke hypercomplex arithmetic (Sangwine et al., 2010). In modern machine learning, explicit incorporation of such symmetries through complex architectures promotes invariance, data efficiency, and generalization (Lattner et al., 2019, Honor et al., 3 Apr 2026). In quantum theory and quantum-classical simulations, basis flexibility directly translates into optimal resource utilization and physical transparency (Miyazaki et al., 2024, Boto et al., 2020).

A plausible implication is that, as algebraic and computational frameworks continue to accommodate more general forms of complex-valued transformations (including those in hypercomplex or multi-field models), future advances will lie in designing basis-invariant, phase-covariant, or internally symmetric algorithms and representations, both for theoretical classification and for efficient, robust computation.