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arXiv:1704.03199 · Physical Review Letters

Monogamy Inequality for Any Local Quantum Resource and Entanglement

S. Camalet

Published Sep 14, 2017• 29 citations
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Abstract

We derive a monogamy inequality for any local quantum resource and entanglement. It results from the fact that there is always a convex measure for a quantum resource, as shown here, and from the relation between entanglement and local entropy. One of its consequences is an entanglement monogamy different from that usually discussed. If the local resource is nonuniformity or coherence, it is satisfied by familiar resource and entanglement measures. The ensuing upper bound for the local coherence, determined by the entanglement, is independent of the basis used to define the coherence.

DOI
10.1103/physrevlett.119.110503

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We derive a monogamy inequality for any local quantum resource and entanglement. It results from the fact that there is always a convex measure for a quantum resource, as shown here, and from the relation between entanglement and local entropy. One of its consequences is an entanglement monogamy different from that usually discussed. If the local resource is nonuniformity or coherence, it is satisfied by familiar resource and entanglement measures. The ensuing upper bound for the local coherence, determined by the entanglement, is independent of the basis used to define the coherence.
We derive a monogamy inequality for entanglement and local contextuality, for any finite bipartite system. It essentially results from the relations between the entropy of a local state and the entanglement of the global state, and between the purity of a state, in the sense of majorization, and its ability to violate a given state-dependent noncontextuality inequality. We build an explicit entanglement monotone that satisfies the found monogamy inequality. An important consequence of this inequality is that there are global states too entangled to violate the local noncontextuality inequality.
We establish duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, as quantified by linear entropy, we prove that the same quantity (namely, linear entropy) provides a lower bound for distribution of bipartite entanglement in a multipartite system. Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest.
One fundamental feature of quantum entanglement is the monogamy which shows that the entanglement of two systems limits the ability of either system to entangle with a third one. Such an understanding is only well described in the qubit systems, but it remains an open question for high-dimensional quantum systems. Here the relative entropy of entanglement and the negativity are respectively used to quantify the entanglement. Based on the resource theory of coherence, we found that the monogamy inequalities in arbitrarily finite-dimensional systems can be successfully established for the entanglement induced by quantum coherence. Moreover, the similar inequalities are also constructed for the distribution of quantum coherence.
Abstract In the present study, a trade off of sharing of entanglement between subsystems of a higher dimensional quantum state is derived. It is presented in terms of an inequality which is analogous to the Coffman–Kundu–Wootters inequality that succinctly describes monogamy of entanglement in dimensional pure state. To derive the monogamy inequality in dimension, G‐concurrence measure of entanglement is considered as a measure of entanglement of maximal dimension. The approach of the present study incidentally points toward a rigorous framework which enables to obtain an upper bound of G‐concurrence of a bipartite qudit mixed state. Obtained upper bound of G‐concurrence is then shown to satisfy a monogamy relation.
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.
One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the αth power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems for α≥4ln2. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0≤β≤2. Given that the logarithmic negativity and LCREN are not convex, these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems in terms of the αth power of logarithmic negativity and LCREN for α≥4ln2. We also show that the βth power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for 0≤β≤2.