We consider a single copy of a pure four-partite state of qubits and investigate its behaviour under the action of stochastic local quantum operations assisted by classical communication (SLOCC). This leads to a complete classification of all different classes of pure states of four-qubits. It is shown that there exist nine families of states corresponding to nine different ways of entangling four qubits. The states in the generic family give rise to GHZ-like entanglement. The other ones contain essentially 2- or 3-qubit entanglement distributed among the four parties. The concept of concurrence and 3-tangle is generalized to the case of mixed states of 4 qubits, giving rise to a seven parameter family of entanglement monotones. Finally, the SLOCC operations maximizing all these entanglement monotones are derived, yielding the optimal single copy distillation protocol

We translate the action of local Clifford operations on graph states into transformations on their associated graphs -i.e. we provide transformation rules, stated in purely graph theoretical terms, which completely characterize the evolution of graph states under local Clifford operations. As we will show, there is essentially one basic rule, successive application of which generates the orbit of any graph state under local unitary operations within the Clifford group.

A general mathematical framework is presented to describe local equivalence classes of multipartite quantum states under the action of local unitary and local filtering operations. This yields multipartite generalizations of the singular value decomposition. The analysis naturally leads to the introduction of entanglement measures quantifying the multipartite entanglement (as generalizations of the concurrence and the 3-tangle), and the optimal local filtering operations maximizing these entanglement monotones are obtained. Moreover a natural extension of the definition of GHZ-states to e.g. $2\times 2\times N$ systems is obtained.Comment: Proof of uniqueness of normal form adde

We consider one single copy of a mixed state of two qubits and investigate how its entanglement changes under local quantum operations and classical communications (LQCC) of the type $\rho'\sim (A\otimes B)\rho(A\otimes B)^{\dagger}$. We consider a real matrix parameterization of the set of density matrices and show that these LQCC operations correspond to left and right multiplication by a Lorentz matrix, followed by normalization. A constructive way of bringing this matrix into a normal form is derived. This allows us to calculate explicitly the optimal local filterin operations for concentrating entanglement. Furthermore we give a complete characterization of the mixed states that can be purified arbitrary close to a Bell state. Finally we obtain a new way of calculating the entanglement of formation.Comment: 4 page

In this paper we investigate two different entanglement measures in the case of mixed states of two qubits. We prove that the negativity of a state can never exceed its concurrence and is always larger then (1 − C) 2 + C 2 − (1 − C) where C is the concurrence of the state. Furthermore we derive an explicit expression for the states for which the upper or lower bound is satisfied. Finally we show that similar results hold if the relative entropy of entanglement and the entanglement of formation are compared. 03.65.BzThe concept of negativity originates from the observation due to Peres [1] that taking a partial transpose of a density matrix associated with a separable state is still a valid density matrix and thus positive (semi)definite. Subsequently M.Horodecki,P.Horodecki and R.Horodecki [2] proved that this was a necessary and sufficient condition for a state to be separable if the dimension of the Hilbert space does not exceed 6. In the case of an entangled mixed state two qubits, the negativity is defined as two times the absolute values of the negative eigenvalue of the partial transpose of a state. Recently, Vidal and Werner proved that the negativity is an entanglement monotone and therefore a good entanglement measure [3]. Furthermore, the concept of negativity is of importance as it leads to upper bounds for the entanglement of distillation.The concept of concurrence originates from the seminal work of Hill and Wootters [4,5] where the exact expression of the entanglement of formation of a system of two qubits was derived. They showed that the entanglement of formation, an entropic entanglement monotone, is a convex monotonic increasing function of the concurrence.Both measures have the same dimensionality and it is therefore a natural question to compare them, as one is related to the concept of entanglement of formation and the other one to the concept of entanglement of distillation.We will derive the possible range of values for the negativity if the concurrence of the state is known. First of all we prove the following conjecture by Eisert and Plenio [8]:Theorem 1 The negativity of an entangled mixed state of two qubits can never exceed its concurrence.To prove this, we need the result of Wootters [5] that a state with a given concurrence can always be decomposed as a convex sum of four pure states all having the same concurrence. It is readily checked that the negativity of a pure state is exactly equal to its concurrence. Due to linearity of the partial trace operation, the negativity of a mixed state is now obtained by calculating the smallest eigenvalue of the matrix obtained by making the convex sum of the partial transposes of the four pure states which have all an equal negative eigenvalue. It is a well-known result due to Weyl that the minimal eigenvalue of the sum of matrices always exceeds the sum of the minimal eigenvalues, which concludes the proof.2The next step is to find the lowest possible value of the negativity for given concurrence. To this end we need a parameterization of the ...