An SL -invariant extension of the concurrence to higher local Hilbert-space dimension is due to its relation with the determinant of the matrix of a d × d two qudits state, which is the only SL -invariant of polynomial degree d . This determinant is written in terms of antilinear expectation values of the local SL ( d ) operators. We use the permutation invariance of the comb-condition for creating further local antilinear operators which are orthogonal to the original operator. It means that the symmetric group acts transitively on the space of combs of a given order. This extends the mechanism for writing -invariants for qubits to qudits. I outline the method, which in principle works for arbitrary dimension d , explicitly for spin 1 and spin 3/2. There is an odd–even discrepancy: whereas for half odd integer spin a situation similar to that observed for qubits is found, for integer spin the outcome is an asymmetric invariant of polynomial degree 2 d .
An SL -invariant extension of the concurrence to higher local Hilbert-space dimension is due to its relation with the determinant of the matrix of a d × d two qudits state, which is the only SL -invariant of polynomial degree d . This determinant is written in terms of antilinear expectation values of the local SL ( d ) operators. We use the permutation invariance of the comb-condition for creating further local antilinear operators which are orthogonal to the original operator. It means that the symmetric group acts transitively on the space of combs of a given order. This extends the mechanism for writing -invariants for qubits to qudits. I outline the method, which in principle works for arbitrary dimension d , explicitly for spin 1 and spin 3/2. There is an odd–even discrepancy: whereas for half odd integer spin a situation similar to that observed for qubits is found, for integer spin the outcome is an asymmetric invariant of polynomial degree 2 d .
The invariant-comb approach is a method to construct entanglement measures for multipartite systems of qubits. The essential step is the construction of an antilinear operator that we call comb in reference to the hairy-ball theorem. An appealing feature of this approach is that, for qubits (or spins 1/2), the combs are automatically invariant under SL(2, C), which implies that the obtained invariants are entanglement monotones by construction. By asking which property of a state determines whether or not it is detected by a polynomial SL(2, C) invariant, we find that it is the presence of a balanced part that persists under local unitary transformations. We present a detailed analysis for the maximally entangled states detected by such polynomial invariants, which leads to the concept of irreducibly balanced states. The latter indicates a tight connection with stochastic local operations and classical communication (SLOCC) classifications of qubit entanglement. Combs may also help to define measures for multipartite entanglement of higher-dimensional subsystems. However, for higher spins there are many independent combs, such that it is nontrivial to find an invariant one. By restricting the allowed local operations to rotations of the coordinate system (i.e. again to the SL(2, C)), we manage to define a unique extension of the concurrence to general half-integer spin with an analytic convex-roof expression for mixed states.
It is a recent observation that entanglement classification for qubits is closely related to local SL(2,C)-invariants including the invariance under qubit permutations [Dür, et al., Phys. Rev. A 62, 062314 (2000); Osterloh, A. and Siewert, J., Phys. Rev. A 72, 012337 (2005); O. Chterental and D. Ž. Ðoković, Linear Algebra Research Advances (Nova Science, Hauppauge, N.Y., 2007), Chap. 4, p. 133], which has been termed SL∗ invariance. In order to single out the SL∗ invariants, we analyze the SL(2,C)-invariants of four (five) qubits and decompose them into irreducible modules for the symmetric group S4 (S5) 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 by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005); Int. J. Quant. Inf. 4, 531 (2006)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the Ω-process [Luque, J.-G. and Thibon, J.-Y., J. Phys. A 39, 371 (2005)]. As the degrees of invariants increase, the alternative method proves to be particularly efficient.
The invariant-comb approach is a method to construct entanglement measures for multipartite systems of qubits. The essential step is the construction of an antilinear operator that we call comb in reference to the hairy-ball theorem . An appealing feature of this approach is that, for qubits (or spins 1/2), the combs are automatically invariant under , which implies that the obtained invariants are entanglement monotones by construction. By asking which property of a state determines whether or not it is detected by a polynomial invariant, we find that it is the presence of a balanced part that persists under local unitary transformations. We present a detailed analysis for the maximally entangled states detected by such polynomial invariants, which leads to the concept of irreducibly balanced states. The latter indicates a tight connection with stochastic local operations and classical communication (SLOCC) classifications of qubit entanglement. Combs may also help to define measures for multipartite entanglement of higher-dimensional subsystems. However, for higher spins there are many independent combs, such that it is nontrivial to find an invariant one. By restricting the allowed local operations to rotations of the coordinate system (i.e. again to the SL ), we manage to define a unique extension of the concurrence to general half-integer spin with an analytic convex-roof expression for mixed states.
We present a method to construct entanglement measures for pure states of multipartite qubit systems. The key element of our approach is an antilinear operator that we call comb in reference to the hairy-ball theorem. For qubits (i.e. spin 1/2) the combs are automatically invariant under SL (2, ℂ). This implies that the filters obtained from the combs are entanglement monotones by construction. We give alternative formulae for the concurrence and the 3-tangle as expectation values of certain antilinear operators. As an application we discuss inequivalent types of genuine four-, five- and six-qubit entanglement.
We present a method to construct entanglement measures for pure states of multipartite qubit systems. The key element of our approach is an antilinear operator that we call ``comb''. For qubits (or spin 1∕2) the combs are automatically invariant under SL(2,C). This implies that the filters obtained from the combs are entanglement monotones by construction. We give alternative formulas for the concurrence and the three-tangle as expectation values of certain antilinear operators. As an application we discuss inequivalent types of genuine four-qubit entanglement.
We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.