Bipartite Mixed States Research Articles

Some aspects of separability

We present three necessary separability criteria for bipartite mixed states; the violation of each of these conditions is a sufficient condition for entanglement. Some ideas on the issue of finding a necessary and sufficient criterion of separability are also discussed.

Purification and correlated measurements of bipartite mixed states

We prove that all purifications of a non-factorable state (i.e., the state which cannot be expressed in a form $\rho_{AB}=\rho_A\otimes\rho_B$) are entangled. We also show that for any bipartite state there exists a pair of measurements which are correlated on this state if and only if the state is non-factorable.

Irreversibility in asymptotic manipulations of a distillable entangled state

We provide an example of a distillable bipartite mixed state such that, even in the asymptotic limit, more pure-state entanglement is required to create it than can be distilled from it. Thus, we show that the irreversibility in the processes of formation and distillation of bipartite states, recently proved in [G. Vidal and J. I. Cirac, Phys. Rev. Lett. 86, 5803 (2001)], is not limited to bound-entangled states.

Concurrence in arbitrary dimensions

We argue that a complete characterization of quantum correlations in bipartite systems of many dimensions may require a quantity which, even for pure states, does not reduce to a single number. Subsequently, we introduce multidimensional generalizations of concurrence and find evidence that they may provide useful tools for the analysis of quantum correlations in mixed bipartite states. We also introduce biconcurrence that leads to a necessary and sufficient condition for separability.

Catalysis of entanglement manipulation for mixed states

We consider entanglement-assisted remote quantum state manipulation of bipartite mixed states. Several aspects are addressed: we present a class of mixed states of rank two that can be transformed into another class of mixed states under entanglement-assisted local operations with classical communication, but for which such a transformation is impossible without assistance. Furthermore, we demonstrate enhancement of the efficiency of purification protocols with the help of entanglement-assisted operations. Finally, transformations from one mixed state to mixed target states which are sufficiently close to the source state are contrasted with similar transformations in the pure-state case.

Evidence for bound entangled states with negative partial transpose

We exhibit a two-parameter family of bipartite mixed states $\rho_{bc}$, in a $d\otimes d$ Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in $2\otimes 2$ can be distilled by local quantum operations and classical communication (LQ+CC). Evidence for this undistillability is provided by the result that, for certain states in this family, we cannot extract entanglement from any arbitrarily large number of copies of $\rho_{bc}$ using a projection on $2\otimes 2$. These states are canonical NPT states in the sense that any bipartite mixed state in any dimension with NPT can be reduced by LQ+CC operations to an NPT state of the $\rho_{bc}$ form. We show that the main question about the distillability of mixed states can be formulated as an open mathematical question about the properties of composed positive linear maps.

Optimal decompositions of barely separable states

Two families of bipartite mixed quantum states are studied for which it is proved that the number of members in the optimal-decomposition ensemble—the ensemble realizing the entanglement of formation—is greater than the rank of the mixed state. We find examples for which the number of states in this optimal ensemble can be larger than the rank by an arbitrarily large factor. In one case the proof relies on the fact that the partial transpose of the mixed state has zero eigenvalues; in the other case the result arises from the properties of product bases that are completable only by embedding in a larger Hilbert space.

Unextendible Product Bases and Bound Entanglement

An unextendible product basis (UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled state. We exhibit a tripartite 2x2x2 UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e. all three corresponding 2x4 bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local POVMs and classical communication.