We build the counterpart of the celebrated Nielsen's theorem for coherence manipulation in this paper. This offers an affirmative answer to the open question: whether, given two states $\rho$ and $\sigma$, either $\rho$ can be transformed into $\sigma$ or vice versa under incoherent operations [Phys. Rev. Lett. \textbf{113}, 140401(2014)]. As a consequence, we find that there exist essentially different types of coherence. Moreover, incoherent operations can be enhanced in the presence of certain coherent states. These extra states are coherent catalysts: they allow uncertain incoherent operations to be realized, without being consumed in any way. Our main result also sheds a new light on the construction of coherence measures.Comment: arXiv admin note: text overlap with arXiv:1311.0275 by other authors. 03.65.Ud, 03.67.Ta. in Physical Review A 201

We provide a fine-grained definition of monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms of an equality, as opposed to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.Monogamy of entanglement is one of the nonintuitive phenomena of quantum physics that distinguish it from classical physics. Classically, three random bits can be maximally correlated. For example, three coins can be prepared in a state in which with 50% chance all three coins show "head", and with the other 50% chance they all show "tail".

Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in literatures are not complete. We establish here a strict framework for defining multipartite entanglement measure (MEM): apart from the postulates of bipartite measure [i.e., vanishing on separable and nonincreasing under local operations and classical communication (LOCC)], a genuine MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula under the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, the convex-roof extension of negativity and negativity, respectively. We show that multipartite extensions of the bipartite measures that are defined by the convex-roof structure are completely monogamous, the extensions of EoF, concurrence and tangle are genuine MEMs (an MEM is called a genuine MEM if it satisfies both the unification condition and the hierarchy condition), and multipartite extensions of both negativity and the convex-roof extension of negativity are unified MEMs but not genuine MEMs. PACS numbers: 03.67.Mn, 03.65.Db, 03.65.Ud.Introduction.-Entanglement is recognized as the most important resource in quantum information processing tasks [1]. A fundamental problem in this field is to quantify entanglement. Many entanglement measures have been proposed for this purpose, such as the distillable entanglement [2], entanglement cost [2, 3], entanglement of formation [3, 4], concurrence [5][6][7], tangle [8], relative entropy of entanglement [9,10], negativity [11,12], geometric measure [13][14][15], squashed entanglement [16,17], the conditional entanglement of mutual information [18], three-tangle [19], the generalizations of concurrence [20,21], and the α-entanglement entropy [22], etc. However, apart from the α-entanglement entropy, all other measures are either only defined on the bipartite case or just discussed with only the axioms of the bipartite case.

Quantum discord is a manifestation of quantum correlations due to non-commutativity rather than entanglement. Two measures of quantum discord by the amount of non-commutativity via the trace norm and the Hilbert-Schmidt norm respectively are proposed in this paper. These two measures can be calculated easily for any state with arbitrary dimension. It is shown by several examples that these measures can reflect the amount of the original quantum discord.

The unextendible product basis (UPB) is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number k (UEBk) for any bipartite system