Algebraic structures of the Lindblad equation
4 affiliations
- 1Cinvestav Unidad Zacatenco
- 2Universidad Autónoma Metropolitana
- 3Universidad Autonoma Metropolitana
- 4Universidad Autónoma Metropolitana Azcapotzalco
Abstract
We investigate the algebraic structure underlying the Lindblad
equation for finite-dimensional open quantum systems.
Using an orthonormal Hermitian operator basis, we represent
the Liouville superoperator in a model-independent basis of
Hermitian superoperators realizing the standard left--right
multiplication maps.
We express their multiplication, commutator, and anticommutator
relations explicitly in terms of the Lie and Jordan structure
constants of the underlying operator basis.
This formulation makes explicit how dissipative contributions
can extend the generator beyond the adjoint Lie algebra
associated with purely Hamiltonian evolution.
The multiplication rules allow us to derive differential
equations for the expansion coefficients of the dynamical map.
We further derive recursion relations for the iterative
construction of the algebra in multiqubit systems.
For a fixed operator basis, the algebraic data can be precomputed
and reused, while the Hamiltonian and dissipative parameters
enter through a single set of coefficients.
Computational benchmarks compare memory usage and construction
time for algebraic, direct, and standard Kronecker implementations.
For the cases tested, the algebraic implementation reduces peak
memory requirements and construction time relative to the direct
implementation.
\blue{The standard Kronecker implementation requires less memory and
time than both alternatives and enables calculations for larger
numbers of qubits.}
These findings establish the computational limitations of the
present algebraic implementation alongside its explicit
characterization of the underlying superoperator algebra.
An accompanying Mathematica notebook provides five examples
covering algebraic verification for one-, two-, and three-qubit
operator bases and numerical transmon dynamics.
The examples confirm agreement with the direct method,
illustrate an approximate X gate and population leakage
beyond the computational subspace, and use Choi matrices
to check Hermiticity preservation, complete positivity,
and trace preservation.
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BibTeX
@article{bixano2026algebraic,
title = {{Algebraic structures of the Lindblad equation}},
author = {Leonel De La Cruz Bixano and Guillermo López Alvarez and Victor A Cruz-Barriguete and V.G. Ibarra-Sierra and José Luis Cardoso Cortés and J.C. Sandoval-Santana and Alejandro Kunold},
journal = {New Journal of Physics},
year = {2026},
month = oct,
doi = {10.1088/1367-2630/aebbe5},
url = {https://iopscience.iop.org/article/10.1088/1367-2630/aebbe5},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Algebraic structures of the Lindblad equation
AU - Leonel De La Cruz Bixano
AU - Guillermo López Alvarez
AU - Victor A Cruz-Barriguete
AU - V.G. Ibarra-Sierra
AU - José Luis Cardoso Cortés
AU - J.C. Sandoval-Santana
AU - Alejandro Kunold
PY - 2026
DA - 2026/10/30
JO - New Journal of Physics
DO - 10.1088/1367-2630/aebbe5
UR - https://iopscience.iop.org/article/10.1088/1367-2630/aebbe5
AB - We investigate the algebraic structure underlying the Lindblad
equation for finite-dimensional open quantum systems.
Using an orthonormal Hermitian operator basis, we represent
the Liouville superoperator in a model-independent basis of
Hermitian superoperators realizing the standard left--right
multiplication maps.
We express their multiplication, commutator, and anticommutator
relations explicitly in terms of the Lie and Jordan structure
constants of the underlying operator basis.
This formulation makes explicit how dissipative contributions
can extend the generator beyond the adjoint Lie algebra
associated with purely Hamiltonian evolution.
The multiplication rules allow us to derive differential
equations for the expansion coefficients of the dynamical map.
We further derive recursion relations for the iterative
construction of the algebra in multiqubit systems.
For a fixed operator basis, the algebraic data can be precomputed
and reused, while the Hamiltonian and dissipative parameters
enter through a single set of coefficients.
Computational benchmarks compare memory usage and construction
time for algebraic, direct, and standard Kronecker implementations.
For the cases tested, the algebraic implementation reduces peak
memory requirements and construction time relative to the direct
implementation.
\blue{The standard Kronecker implementation requires less memory and
time than both alternatives and enables calculations for larger
numbers of qubits.}
These findings establish the computational limitations of the
present algebraic implementation alongside its explicit
characterization of the underlying superoperator algebra.
An accompanying Mathematica notebook provides five examples
covering algebraic verification for one-, two-, and three-qubit
operator bases and numerical transmon dynamics.
The examples confirm agreement with the direct method,
illustrate an approximate X gate and population leakage
beyond the computational subspace, and use Choi matrices
to check Hermiticity preservation, complete positivity,
and trace preservation.
SN - 1367-2630
ER -