arXiv subjects

José Luis Cardoso

Publications and source records attributed to José Luis Cardoso.

4 recordsLinked to original sources

arXiv2026

A Lie-Jordan Geometric Formulation of Lindblad Dynamics

We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors $C_{μν}{}^λ$ and $B_{μν}{}^λ$. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, $ \mathcal D(X,Y)Z=XZY-\frac12\{YX,Z\}, $ whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations $(B+C)$ and $(B-C)$ generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.

quant-ph↗

arXiv2026

Algebraic structures of the Lindblad equation

We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.

quant-ph↗

arXiv2021

A $β$-Sturm Liouville problem associated with the general quantum operator

Let $\,I\subseteq\mathbb{R}\,$ be an interval and $\,β:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a unique fixed point $\,s_0\in I\,$ which satisfies $\,(s_0-t)(β(t)-t)\geq 0\,$ for all $\,t\in I$, where the equality holds only when $\,t=s_0$. The general quantum operator defined in 2015 by Hamza et al., $\,D_β[f](t):=\displaystyle\frac{f\big(β(t)\big)-f(t)}{β(t)-t}\,$ if $\,t\neq s_0\,$ and $\,D_β[f](s_0):=f^{\prime}(s_0)\,$ if $\,t=s_0,$ generalizes the Jackson $\,q$-operator $\,D_{q}\,$ and also the Hahn $\,(q,ω)$-operator, $\,D_{q,ω}$. Regarding a $β-$Sturm Liouville eigenvalue problem associated with the above operator $\,D_β\,$, we construct the $β-$Lagrange's identity, show that it is self-adjoint in $\,\mathscr{L}_β^2([a,b]),$ and exhibit some properties for the corresponding eigenvalues and eigenfunctions.

math.CA↗

arXiv2018

On Basic Fourier-Bessel Expansions

When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{kν}$ and the "shifted" zeros, $qj_{kν}$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the "shifted" ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the "shifted" zeros. A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.

math.CA↗