Algebraic solutions for $SU(2)\otimes SU(2)$ Hamiltonian eigensystems: generic statistical ensembles and a mesoscopic system applicationAlgebraic solutions for $SU(2)\otimes SU(2)$ Hamiltonian eigensystems: generic statistical ensembles and a mesoscopic system application

Alex E. Bernardini, Roldão da Rocha

Solutions of generic $SU(2)\otimes SU(2)$ Hamiltonian eigensystems are obtained through systematic manipulations of quartic polynomial equations. An {\em ansatz} for constructing separable and entangled eigenstate basis, depending on the quartic equation coefficients, is proposed. Besides the quantum concurrence for pure entangled states, the associated thermodynamic statistical ensembles, their partition function, quantum purity and quantum concurrence are shown to be straightforwardly obtained. Results are specialized to a $SU(2)\otimes SU(2)$ structure emulated by lattice-layer degrees of freedom of the Bernal stacked graphene, in a context that can be extended to several mesoscopic scale systems for which the onset from $SU(2)\otimes SU(2)$ Hamiltonians has been assumed.Solutions of generic $SU(2)\otimes SU(2)$ Hamiltonian eigensystems are obtained through systematic manipulations of quartic polynomial equations. An {\em ansatz} for constructing separable and entangled eigenstate basis, depending on the quartic equation coefficients, is proposed. Besides the quantum concurrence for pure entangled states, the associated thermodynamic statistical ensembles, their partition function, quantum purity and quantum concurrence are shown to be straightforwardly obtained. Results are specialized to a $SU(2)\otimes SU(2)$ structure emulated by lattice-layer degrees of freedom of the Bernal stacked graphene, in a context that can be extended to several mesoscopic scale systems for which the onset from $SU(2)\otimes SU(2)$ Hamiltonians has been assumed.

arXiv:2501.06182Jan 10, 20250 citations

李代数相关的时间演化问题SU（2）和Heisenberg—...

肖青, 孙昌璞

东北师大学报：自然科学版Jan 1, 19910 citations

Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits

Vladimir P. Gerdt, A. Khvedelidze, Yu. G. Palii

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices $\mathfrak{P}_ +$ . Since elements of $\mathfrak{P}_ +$ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, $\mathfrak{P}_ + \in \mathbb{R}^{15}$ . We define $\mathfrak{P}_ +$ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices $\mathfrak{P}_ +$ . Since elements of $\mathfrak{P}_ +$ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, $\mathfrak{P}_ + \in \mathbb{R}^{15}$ . We define $\mathfrak{P}_ +$ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.

Physics of Atomic NucleiJun 1, 20116 citations

Chain of interacting SU(2)4 anyons and quantum SU(2) k ×$\overline {SU(2)_k } $ doubles

V. A. Verbus, L. Martina, A. P. Protogenov

Theoretical and Mathematical PhysicsJun 1, 20114 citations

二能级原子SU（2）压缩的混态临界性质

刘世炳

Jan 1, 19920 citations

Algebraic solutions for

SU(2)\otimes SU(2)

Hamiltonian eigensystems: generic statistical ensembles and a mesoscopic system application

Solutions of generic

SU(2)\otimes SU(2)

Hamiltonian eigensystems are obtained through systematic manipulations of quartic polynomial equations. An {\em ansatz} for constructing separable and entangled eigenstate basis, depending on the quartic equation coefficients, is proposed. Besides the quantum concurrence for pure entangled states, the associated thermodynamic statistical ensembles, their partition function, quantum purity and quantum concurrence are shown to be straightforwardly obtained. Results are specialized to a

SU(2)\otimes SU(2)

structure emulated by lattice-layer degrees of freedom of the Bernal stacked graphene, in a context that can be extended to several mesoscopic scale systems for which the onset from

SU(2)\otimes SU(2)

Hamiltonians has been assumed.

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices

\mathfrak{P}_ +

. Since elements of

\mathfrak{P}_ +

are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset,

\mathfrak{P}_ + \in \mathbb{R}^{15}

. We define

\mathfrak{P}_ +

explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.