Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

A decomposition theorem for finite-dimensional Hermitian projections

M.J. Giannoni

Published Feb 1, 1981• 2 citations
Publisher

Abstract

We investigate conditions for the existence of a decomposition of a Hermitian projector ρ into two Hermitian and time-reversal invariant operators ρ0 and χ under the form ρ=eiχ ρ0 e−iχ. Sufficient conditions are given and an explicit construction of a decomposition is performed when they are fulfilled. A stronger theorem of existence and uniqueness is studied.

DOI
10.1063/1.524887

Related papers

Characteristic operators and unitarily invariant decomposition of hermitian operators

Auchin Tang, Hong Guo

Abstract A set of characteristic operators {F } is proposed for performing the decomposition of p‐particle Hermitian operators {Dp} to constitute irreducible components {D } of the unitary group D = F Dp, q = 0,1,2,…,p. For a deeper expolration of the properties of the characteristic operators, a few theorems are presented. As an illustration, the expected values for symmetric p‐particle Hermitian operators are obtained as a number of terms having invariant group‐theoretical meaning.

International Journal of Quantum Chemistry•Jan 1, 1983•38 citations

A constructive index theorem, phase operators, and phase-difference operators

Sixia Yu, Yongde Zhang

A constructive reversed index theorem concerning the polar decomposition of an operator into a product of a unitary exponential phase operator and a Hermitian amplitude operator is investigated. Its applications to the construction of Hermitian phase operators of fermions and phase-difference operators between bosons or fermions are presented. Specifically, a Hermitian operator is constructed for the phase difference between a single-mode fermion and boson, which is justified by the fact that it is exactly the interaction part of the Hamiltonian of the Jaynes-Cummings Model. Furthermore, a Hermitian phase operator of a single-mode boson can also be defined referring to a single-mode fermion. All those quantized phases and phase-differences are found to obey a quantum addition rule instead of the ordinary commutative addition rule.

Journal of Mathematical Physics•Oct 1, 1998•7 citations

Lie algebra and unitarily invariant decomposition of Hermitian operators

Chia‐Chung Sun, Xue‐Qui Li, Auchin Tang

Abstract The unitarily invariant decomposition of Hermitian operators is performed by means of Lie algebra.

International Journal of Quantum Chemistry•Apr 1, 1984•15 citations

Thermodynamic Orthogonality from Cartan Decomposition: Coherence Work, Entropy Production, and Their Separation in Open Quantum Systems

Mitsutoshi Yamada

The Cartan decomposition su(d) = h ⊕ n encodes a fundamental thermodynamic separation: coherence work W* = δQ_std − δQ_ent (eigenvector-projector rotation) responds exclusively to root-space perturbations (F_α ∈ n), while the entropy-production gap g responds exclusively to Cartan perturbations (F_β ∈ h). This thermodynamic orthogonality is a structural principle from which three results follow non-perturbatively and without secular conditions. Theorem O (thermodynamic orthogonality): The sub-Jacobian of ρ ↦ (W*, g) at ρ₀ = I/d is block-antidiagonal for any d ≥ 2, with det J_sub ≠ 0 iff the dissipator is non-isotropic. Theorem C (irreducibility, from Thm. O): No f: ℝ → ℝ satisfies δQ_ent + f(W*) = T dS_vN for all ρ. Coherence work is an irreducible thermodynamic quantity. Theorem N (second law without secular approximation): σ_new = −Tr[D[ρ](log ρ − log ρ_ss)] ≥ 0 for any GKSL generator; σ_new is the Cartan-orthogonality-invariant measure of irreversibility, with W*/T = σ_ent − σ_std experimentally accessible. Theorem H_d (trivial case of Thm. O): det J_sub = 0, equivalently W* = 0 for all ρ, if and only if A = c I_{d²−1}, b = 0 (T₁ = T₂ for d = 2) — the isotropic dissipator, the unique case in which the Cartan structure imposes no thermodynamic separation. In summary, Theorem O yields three corollaries: (I) W* is irreducible (Theorem C); (II) a non-negative entropy production σ_new ≥ 0 exists within the GKSL framework, without secular conditions (Theorem N); (III) W* = 0 for all ρ iff T₁ = T₂ (Theorem H_d).

•Jun 14, 2026•0 citations

A Quantum Algorithm for the Radical of a Lie Algebra: Kernel Projection and Conditioning

Yibin Wang

Every finite-dimensional real or complex Lie algebra has a largest solvable ideal, its radical. In the compact dynamical Lie algebra (DLA) of a closed quantum system, this radical is the center; projecting onto it isolates directions that commute with the supplied algebra. With sparse Lie-bracket data and a known spectral gap, we construct a quantum circuit that approximately encodes this coefficient-space projector. The general construction combines the derived-algebra map with the Killing form, whereas compactness makes the Killing form alone sufficient. This distinction matters for numerical sensitivity, and our main theorem gives its exact law under an invariant-orthonormal basis adapted to the center and semisimple part. In a compact real algebra with a nonzero semisimple part, the condition number of the general operator is the three-halves power of the Killing-form condition number on that part. The compact projector in turn yields a bounded-error test for whether the center is trivial, provided that any nonzero center has at least a stated minimum dimension. This gives a controlled test for internal conserved directions in compact quantum dynamics.

quant-pharXiv:2609.02084•Sep 3, 2026•0 citations

On the unitarily invariant decomposition of Hermitian operators

Chia‐Chung Sun, Xue‐Qui Li, Auchin Tang

Abstract The unitarily invariant decomposition of Hermitian operators for a boson system is performed by means of the symmetry of group SO(2, 1) and the decomposition is also closely related with Lie algebra.

International Journal of Quantum Chemistry•Jun 1, 1984•7 citations
We investigate conditions for the existence of a decomposition of a Hermitian projector ρ into two Hermitian and time-reversal invariant operators ρ0 and χ under the form ρ=eiχ ρ0 e−iχ. Sufficient conditions are given and an explicit construction of a decomposition is performed when they are fulfilled. A stronger theorem of existence and uniqueness is studied.
A constructive reversed index theorem concerning the polar decomposition of an operator into a product of a unitary exponential phase operator and a Hermitian amplitude operator is investigated. Its applications to the construction of Hermitian phase operators of fermions and phase-difference operators between bosons or fermions are presented. Specifically, a Hermitian operator is constructed for the phase difference between a single-mode fermion and boson, which is justified by the fact that it is exactly the interaction part of the Hamiltonian of the Jaynes-Cummings Model. Furthermore, a Hermitian phase operator of a single-mode boson can also be defined referring to a single-mode fermion. All those quantized phases and phase-differences are found to obey a quantum addition rule instead of the ordinary commutative addition rule.
The Cartan decomposition su(d) = h ⊕ n encodes a fundamental thermodynamic separation: coherence work W* = δQ_std − δQ_ent (eigenvector-projector rotation) responds exclusively to root-space perturbations (F_α ∈ n), while the entropy-production gap g responds exclusively to Cartan perturbations (F_β ∈ h). This thermodynamic orthogonality is a structural principle from which three results follow non-perturbatively and without secular conditions. Theorem O (thermodynamic orthogonality): The sub-Jacobian of ρ ↦ (W*, g) at ρ₀ = I/d is block-antidiagonal for any d ≥ 2, with det J_sub ≠ 0 iff the dissipator is non-isotropic. Theorem C (irreducibility, from Thm. O): No f: ℝ → ℝ satisfies δQ_ent + f(W*) = T dS_vN for all ρ. Coherence work is an irreducible thermodynamic quantity. Theorem N (second law without secular approximation): σ_new = −Tr[D[ρ](log ρ − log ρ_ss)] ≥ 0 for any GKSL generator; σ_new is the Cartan-orthogonality-invariant measure of irreversibility, with W*/T = σ_ent − σ_std experimentally accessible. Theorem H_d (trivial case of Thm. O): det J_sub = 0, equivalently W* = 0 for all ρ, if and only if A = c I_{d²−1}, b = 0 (T₁ = T₂ for d = 2) — the isotropic dissipator, the unique case in which the Cartan structure imposes no thermodynamic separation. In summary, Theorem O yields three corollaries: (I) W* is irreducible (Theorem C); (II) a non-negative entropy production σ_new ≥ 0 exists within the GKSL framework, without secular conditions (Theorem N); (III) W* = 0 for all ρ iff T₁ = T₂ (Theorem H_d).
Every finite-dimensional real or complex Lie algebra has a largest solvable ideal, its radical. In the compact dynamical Lie algebra (DLA) of a closed quantum system, this radical is the center; projecting onto it isolates directions that commute with the supplied algebra. With sparse Lie-bracket data and a known spectral gap, we construct a quantum circuit that approximately encodes this coefficient-space projector. The general construction combines the derived-algebra map with the Killing form, whereas compactness makes the Killing form alone sufficient. This distinction matters for numerical sensitivity, and our main theorem gives its exact law under an invariant-orthonormal basis adapted to the center and semisimple part. In a compact real algebra with a nonzero semisimple part, the condition number of the general operator is the three-halves power of the Killing-form condition number on that part. The compact projector in turn yields a bounded-error test for whether the center is trivial, provided that any nonzero center has at least a stated minimum dimension. This gives a controlled test for internal conserved directions in compact quantum dynamics.