Journal of Mathematical Sciences

Constructing the SU(2) × U(1) Orbit Space for Qutrit Mixed States

Vladimir P. Gerdt, A. Khvedelidze, Yuri Palii

Published Aug 17, 2015• 1 citation

Abstract

The orbit space of the group G := SU(2) × U(1) ⊂ U(3) acting adjointly on the state space of a three-level quantum system is discussed. The semialgebraic structure of is determined within the Procesi–Schwarz method. Using an integrity basis for the ring of G-invariant polynomials , the set of constraints on the Casimir invariants of the group U(3) coming from the positivity requirement Grad(z) ≥ 0 for the Procesi–Schwarz gradient matrix is analyzed in detail.

DOI: 10.1007/s10958-015-2535-x

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Journal of Mathematical Sciences

Constructing the SU(2) × U(1) Orbit Space for Qutrit Mixed States

Vladimir P. Gerdt, A. Khvedelidze, Yuri Palii

Published Aug 17, 2015• 1 citation

Publisher

Abstract

DOI: 10.1007/s10958-015-2535-x

Related papers

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The orbit space

\mathfrak{P}(\mathbb{R}^8)/\mathrm{G}

, of the group

\mathrm{G}:=\mathrm{SU(2)\times U(1)}\subset\mathrm{U(3)}

acting adjointly on the state space

\mathfrak{P}(\mathbb{R}^8)

of a 3-level quantum system is discussed. The semi-algebraic structure of

\mathfrak{P}(\mathbb{R}^8) /\mathrm{G}

, is determined within the Procesi-Schwarz method. Using the integrity basis for the ring of G-invariant polynomials,

\mathbb{R}[\mathfrak{P}(\mathbb{R}^8)]^{\mathrm{G}}

, the set of constraints on the Casimir invariants of

\mathrm{U}(3)

group coming from the positivity requirement of Procesi-Schwarz gradient matrix,

\mathrm{Grad}(z)\geqslant 0

, is analyzed in details.

The unitary $ \mathrm{U}(d)$-equivalence relation between elements of the space

\mathfrak{P}_+\,

of mixed states of

d

-dimensional quantum system defines the orbit space $ \mathfrak{P}_+/ \mathrm{U}(d)\,$ and provides its description in terms the ring

\mathbb{R}[\mathfrak{P}_+]^{\mathrm{U}(d)}\,

\mathrm{U}(d)

-invariant polynomials. We prove that the semi-algebraic structure of $ \mathfrak{P}_+/ \mathrm{U}(d)\, $ is determined completely by two basic properties of density matrices, their semi-positivity and Hermicity. Particularly, it is shown that the Processi-Schwarz inequalities in elements of integrity basis for

\mathbb{R}[\mathfrak{P}_+]^{\mathrm{U}(d)}\,

defining the orbit space, are identically satisfied for all elements of

\mathfrak{P}_+

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.

The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group

SL(3,{\mathbb {C}})