We reduce the question of whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure states constituting a decomposition of the considered mixture. The method reproduces many known entanglement and/or separability criteria, and provides yet another geometrical characterization of mixed separable states.
We reduce the question of whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure states constituting a decomposition of the considered mixture. The method reproduces many known entanglement and/or separability criteria, and provides yet another geometrical characterization of mixed separable states.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
We study families of positive and completely positive maps acting on a bipartite system ${\mathbb{C}}^{M}\ensuremath{\bigotimes}{\mathbb{C}}^{N}$ (with $M\ensuremath{\le}N$). The maps have a property that, when applied to any state (of a given entanglement class), result in a separable state or, more generally, a state of another certain entanglement class (e.g., Schmidt number $\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary M,N, with a special emphasis on M=2. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to ``weakly'' entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure states, partial transposes, and/or local transformations thereof.
Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] — basing on realignment of elements of density matrix. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria.
The separability from spectrum problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU † is separable for all unitary matrices U. We provide evidence that ρ is separable from spectrum if and only if UρU † has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest. 1
We present a method to derive separability criteria for different classes of multiparticle entanglement, especially genuine multiparticle entanglement. The resulting criteria are necessary and sufficient for certain families of states. This, for example, completely solves the problem of classifying N -qubit Greenberger–Horne–Zeilinger states mixed with white noise according to their separability and entanglement properties. Further, the criteria are superior to all known entanglement criteria for many other families; also they allow the detection of bound entanglement. We next demonstrate that they are easily implementable in experiments and discuss applications to the decoherence of multiparticle entangled states.
In quantum systems, entanglement corresponds to nonclassical correlation of nonlocal observables. Thus, entanglement (or, to the contrary, separability) of a given quantum state is not uniquely determined by properties of the state, but may depend on the choice of the factorization of the algebra of observables. In the present work, we expose and systematize some recently reported results about the possibility to represent a single quantum state as either entangled or separable. We will distinguish in particular the cases of pure and mixed states. For pure states, it has been shown that observables can always be constructed such that any state has any amount of entanglement possible. For mixed states, the situation is more complex and only partial results are known: while it is always possible to choose a factorization such that a state appears separable, a general criterion to determine whether a state can be represented as entangled is not known. These results will be illustrated by several examples, the phenomenon of quantum teleportation, and the geometry of the states of two qubits.