The invariant-comb approach and its relation to the balancedness of multipartite entangled states

TL;DR

The paper introduces the invariant-comb approach for constructing entanglement measures in multipartite qubit systems, linking balanced parts of states to polynomial invariants and SLOCC classifications, and extends these ideas to higher spins.

Contribution

It develops the invariant-comb method for entanglement detection, analyzes maximally entangled states via irreducibly balanced states, and extends concurrence to higher spins with a convex-roof formula.

Findings

Invariant-comb approach yields entanglement monotones for qubits.

Balanced parts of states determine detection by polynomial invariants.

Extension of concurrence to higher spins with an analytic convex-roof expression.

Abstract

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 {\em comb} in reference to the {\em hairy-ball theorem}. An appealing feature of this approach is that for qubits (or spins 1/2) the combs are automatically invariant under $SL(2,\CC)$, 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 $SL(2,\CC)$ invariant we find that it is the presence of a {\em 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 {\em irreducibly balanced} states. The latter indicates a tight connection with 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 non-trivial to find an invariant one. By restricting the allowed local operations to rotations of the coordinate system (i.e. again to the $SL(2,\CC)$) we manage to define a unique extension of the concurrence to general half-integer spin with an analytic convex-roof expression for mixed states.