Entanglement monotones and maximally entangled states in multipartite qubit systems

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 {\em comb} in reference to the {\em hairy-ball theorem}. For qubits (or spin 1/2) the combs are automatically invariant under SL(2,\CC). This implies that the {\em 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.Comment: 7 pages, revtex4. Talk presented at the Workshop on "Quantum entanglement in physical and information sciences", SNS Pisa, December 14-18, 200

Algebraic invariants of five qubits

The Hilbert series of the algebra of polynomial invariants of pure states of five qubits is obtained, and the simplest invariants are computed.Comment: 4 pages, revtex. Short discussion of quant-ph/0506073 include

Rescaling multipartite entanglement measures for mixed states

A relevant problem regarding entanglement measures is the following: Given an arbitrary mixed state, how does a measure for multipartite entanglement change if general local operations are applied to the state? This question is nontrivial as the normalization of the states has to be taken into account. Here we answer it for pure-state entanglement measures which are invariant under determinant 1 local operations and homogeneous in the state coefficients, and their convex-roof extension which quantifies mixed-state entanglement. Our analysis allows to enlarge the set of mixed states for which these important measures can be calculated exactly. In particular, our results hint at a distinguished role of entanglement measures which have homogeneous degree 2 in the state coefficients.Comment: Published version plus one important reference (Ref. [39]

Concurrence classes for an arbitrary multi-qubit state based on positive operator valued measure

In this paper, we propose concurrence classes for an arbitrary multi-qubit state based on orthogonal complement of a positive operator valued measure, or POVM in short, on quantum phase. In particular, we construct concurrence for an arbitrary two-qubit state and concurrence classes for the three- and four-qubit states. And finally, we construct $W^{m}$ and $GHZ^{m}$ class concurrences for multi-qubit states. The unique structure of our POVM enables us to distinguish different concurrence classes for multi-qubit states.Comment: 8 page

Concurrence classes for general pure multipartite states

We propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator valued measure on quantum phase. In particular, we construct $W^{m}$ class, $GHZ^{m}$, and $GHZ^{m-1}$ class concurrences for general pure $m$-partite states. We give explicit expressions for $W^{3}$ and $GHZ^{3}$ class concurrences for general pure three-partite states and for $W^{4}$, $GHZ^{4}$, and $GHZ^{3}$ class concurrences for general pure four-partite states.Comment: 14 page

Multipartite-entanglement monotones and polynomial invariants

We show that a positive homogeneous function that is invariant under determinant 1 stochastic local operations and classical communication (SLOCC) transformations defines an N-qubit entanglement monotone if and only if the homogeneous degree is not larger than four. We then describe a common basis and formalism for the N-tangle and other known invariant polynomials of degree four. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A 67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration, thus giving a physical interpretation for those invariants. We demonstrate that this is a special case of a completely general law that holds for any multipartite system with bipartitions of equal dimension, e.g., for an even number of qudits