In this work, we investigate a two-state system coupled to a generic dissipative environment. Conventionally, such systems are described using a non-Hermitian Hamiltonian within the complexified Pauli matrix algebra, and their dynamics are governed by the non-Hermitian Liouville-von Neumann equation, which is traditionally solved via matrix methods to find the density operator $\hat{\rho}(t)$ . By leveraging the isomorphism between the Pauli matrix algebra $\text{Mat}(2,\mathbb C)$ and the Clifford algebra of three-dimensional space $Cl_3$ , we completely translate the non-Hermitian problem into the geometric framework of $Cl_3$ . This approach natively decouples the dynamics into distinct geometric grades, successfully bypassing the need for cumbersome matrix techniques, such as finding the eigenvalues and eigenvectors of a $4\times 4$ matrix. We demonstrate that the non-unitary time evolution is driven by a generalized geometric spinor that seamlessly unifies the dissipative decay and the Hamiltonian rotation. Finally, we test our general algebraic solution in the Hermitian regime, showing perfect agreement with well-known unitary results.In this work, we investigate a two-state system coupled to a generic dissipative environment. Conventionally, such systems are described using a non-Hermitian Hamiltonian within the complexified Pauli matrix algebra, and their dynamics are governed by the non-Hermitian Liouville-von Neumann equation, which is traditionally solved via matrix methods to find the density operator $\hat{\rho}(t)$. By leveraging the isomorphism between the Pauli matrix algebra $\text{Mat}(2,\mathbb C)$ and the Clifford algebra of three-dimensional space $Cl_3$, we completely translate the non-Hermitian problem into the geometric framework of $Cl_3$. This approach natively decouples the dynamics into distinct geometric grades, successfully bypassing the need for cumbersome matrix techniques, such as finding the eigenvalues and eigenvectors of a $4\times 4$ matrix. We demonstrate that the non-unitary time evolution is driven by a generalized geometric spinor that seamlessly unifies the dissipative decay and the Hamiltonian rotation. Finally, we test our general algebraic solution in the Hermitian regime, showing perfect agreement with well-known unitary results.

Sep 30, 20260 citations

Theory of Density Matrices

K. Husimi, Yoshihiro Kitano, Toshiyuki Nishiyama

Fortschritte der PhysikJan 1, 195827 citations

Clifford Algebras and Geometry of Entanglement

Isamu Imahori

In this thesis, the complex Clifford algebra and its representation are introduced as a foundation of our geometrical description of quantum states. An inner product and a metric are defined on the matrix representation in a natural way, so that it has the structure of a Hilbert space and a Riemannian manifold. In particular, the set of D x D pure density operators which de-scribe n-qubit pure states, where D = 2\$^n\$ and n is a natural number, is characterised as the set of D x D positive operators whose associated quadratic forms satisfy the Fierz identities and the normalisation condition. Then this set is, in fact, nothing but the complex projective space ‚Äövëvá‚Äövëv¥\$^{D-1}\$. Further-more, the fibre bundle S\$^{2D-1}\$ ‚Äövúv¿ ‚Äövëvá‚Äövëv¥\$^{D-1}\$ is derived from the construction of the space of D x D pure density operators, and its bundle projection provides the natural correspondence between the two formulations of quantum mechanics, namely, the state vector formulation and the density operator formulation. The single qubit state case and the two-qubit pure state case are ex-plored intensively as examples. For the single qubit case, both pure and mixed states are discussed explicitly in terms of the Clifford algebra description along with the `mixedness' of a single qubit state, and the Riemannian structure of the space of single qubit pure density operators is examined. For the two-qubit case, the Clifford algebra description is discussed explicitly in relation to the Fierz identities, and, by employing the reduced density operators, it is shown that the concurrence of a two-qubit pure density operator coincides with the 'mixedness' of the reduced density operators. In addition, from this viewpoint, the \EPR paradox\" is examined as a more geometrically precise illustrated example."In this thesis, the complex Clifford algebra and its representation are introduced as a foundation of our geometrical description of quantum states. An inner product and a metric are defined on the matrix representation in a natural way, so that it has the structure of a Hilbert space and a Riemannian manifold. In particular, the set of D x D pure density operators which de-scribe n-qubit pure states, where D = 2\$^n\$ and n is a natural number, is characterised as the set of D x D positive operators whose associated quadratic forms satisfy the Fierz identities and the normalisation condition. Then this set is, in fact, nothing but the complex projective space ‚Äövëvá‚Äövëv¥\$^{D-1}\$. Further-more, the fibre bundle S\$^{2D-1}\$ ‚Äövúv¿ ‚Äövëvá‚Äövëv¥\$^{D-1}\$ is derived from the construction of the space of D x D pure density operators, and its bundle projection provides the natural correspondence between the two formulations of quantum mechanics, namely, the state vector formulation and the density operator formulation. The single qubit state case and the two-qubit pure state case are ex-plored intensively as examples. For the single qubit case, both pure and mixed states are discussed explicitly in terms of the Clifford algebra description along with the `mixedness' of a single qubit state, and the Riemannian structure of the space of single qubit pure density operators is examined. For the two-qubit case, the Clifford algebra description is discussed explicitly in relation to the Fierz identities, and, by employing the reduced density operators, it is shown that the concurrence of a two-qubit pure density operator coincides with the 'mixedness' of the reduced density operators. In addition, from this viewpoint, the \EPR paradox\" is examined as a more geometrically precise illustrated example."

Jan 1, 20080 citations

A new representation for the density matrix

G. Ramachandran, M. V. N. Murthy

Nuclear Physics AJul 1, 197921 citations

\hat{\rho}(t)

. By leveraging the isomorphism between the Pauli matrix algebra

\text{Mat}(2,\mathbb C)

and the Clifford algebra of three-dimensional space

Cl_3

, we completely translate the non-Hermitian problem into the geometric framework of

Cl_3

. This approach natively decouples the dynamics into distinct geometric grades, successfully bypassing the need for cumbersome matrix techniques, such as finding the eigenvalues and eigenvectors of a

4\times 4

matrix. We demonstrate that the non-unitary time evolution is driven by a generalized geometric spinor that seamlessly unifies the dissipative decay and the Hamiltonian rotation. Finally, we test our general algebraic solution in the Hermitian regime, showing perfect agreement with well-known unitary results.