On the ring of local polynomial invariants for a pair of entangled qubits |
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Authors: | V Gerdt A Khvedelidze Yu Palii |
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Institution: | 1.Joint Institute for Nuclear Research,Dubna,Russia;2.Razmadze Mathematical Institute,Tbilisi,Georgia;3.Institute of Applied Physics,Chisinau,Moldova |
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Abstract: | The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of the group SU(2) ⊗ SU(2)
on the space of density matrices
\mathfrakP+ {\mathfrak{P}_{+} } , defined as the space of 4 × 4 positive semidefinite Hermitian matrices. The corresponding ring
\textC \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} of polynomial invariants is studied. A special integrity basis for
\textC \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} is described, and the constraints on its elements imposed by the positive semidefiniteness of density matrices are given
explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that the minimum number
of invariants, namely, two primary invariants of degree 2, 3 and one secondary invariant of degree 4 appearing in the Hironaka
decomposition of
\textC \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} , are subject to the polynomial inequalities. Bibliography: 32 titles. |
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