On polynomial invariants of several qubits

Abstract

It is a recent observation that entanglement classification for qubits is closely related to local $SL(2,\CC)$-invariants including the invariance under qubit permutations, which has been termed $SL^*$ invariance. In order to single out the $SL^*$ invariants, we analyze the $SL(2,\CC)$-invariants of four resp. five qubits and decompose them into irreducible modules for the symmetric group $S_4$ resp. $S_5$ of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of $SL^*$-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced in [Phys. Rev. A 72, 012337 (2005)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the $\Omega$-process. As the degrees of invariants increase, the alternative method proves to be particularly efficient.Comment: 29 pages, 3 eps figures, aipproc. Minor modifications and corrections. Length change only due to style chang

Quantifying entanglement resources

“…With this definition H has a (physically irrelevant) prefactor of 2 compared to the definitions in [198] and [199]. We note that this quantity, in a strict sense, cannot be a measure of genuine entanglement as it yields 1 on a tensor product of two two-qubit Bell states [200].…”

“…There are the degree-8 filter F (4) 2 , and the degree-12 filter F (4) 3 [58,202,199] whose precise definitions are also given in section 6.4.4. The peculiarity of the filters F…”

Classification of Entanglement in Symmetric States

“…As a Platonic state, it is an optimal state for reference frame alignment [106], and in terms of symmetric informationally complete positive-operator-valued measures (SIC POVM) [182,183] it was found that the tetrahedron state is the unique state that can be generated in the setting of a two-dimensional Hilbert space [110,183]. In Section 5.7 it will be outlined that -along with the four qubit cluster state and GHZ state -the tetrahedron state is one of the three maximally entangled four qubit states under a monotone that requires all k-tangles with k < 4 to vanish [58,62].…”

“…For 4 qubits there exist three inequivalent SLOCC invariants 8 , and a corresponding "basis" of three inequivalent maximally 4-tangled states with neither 3-tangle nor concurrence has been determined [58,62]. These states are the GHZ state (|1111〉 + |1100〉 + |0010〉 + |0001〉) .…”

“…For four and five qubits polynomial invariants have also been constructed [57][58][59][60][61][62]. In [57,60] the polynomial invariants were constructed from classical invariant theory, and the values of these invariants for the EFs [51] were derived.…”