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

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

在克尔介质中su（2）相干态光的相位性质

路洪

量子光学学报Jan 1, 19970 citations

Цепочка взаимодействующих $SU(2)_{4}$ -энионов и квантовые $SU(2)_{k}\times\overline{SU(2)_{k}}$ дублиЦепочка взаимодействующих $SU(2)_{4}$-энионов и квантовые $SU(2)_{k}\times\overline{SU(2)_{k}}$ дубли

Валерий Альфонасович Вербус, L. Martina, L. Martina +1 more

Теоретическая и математическая физикаJan 1, 20110 citations

双模SU（2）相干态场与两个耦合原子相互作用的动力学

田永红, 彭金生

Acta Physica SinicaJan 20, 20000 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.

Цепочка взаимодействующих

SU(2)_{4}

-энионов и квантовые

SU(2)_{k}\times\overline{SU(2)_{k}}

дубли