Abstract
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose 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.
Highlights
Monogamy of entanglement is one of the nonintuitive phenomena of quantum physics that distinguish it from classical physics
We provide a characterization for the disentangling condition in the form of an equality between the entanglement of formation (EoF) associated with the given entanglement measure (see Eq (7) below) and the entanglement of assistance (EoA) [58], and discuss its relation to quantum Markov chains [59]
We showed that our notion of monogamy can reproduce monogamy relations like in (1) with a small change that the measure E is replaced by Eα for some exponent α > 0
Summary
Since E is a measure of entanglement, it is non-increasing under partial traces, and E(ρA|BC) ≥ max{E(ρAB), E(ρAC)} for any state ρA|BC ∈ SABC. We denote by f (ρABC) the smallest value of γ that achieves equality in (15). As discussed in the paper, the expression for α in (16) is optimal in the sense that it provides the smallest possible value for α that satisfies Eq (5). This monogamy exponent is a function of the measure E, and we denote it by α(E). Almost all the entanglement measures studied in the literature are continuous, and in particular C, N , Ncr, Ef , τ , Tq, Rα and Er are all continuous [27, 28, 29, 30]
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