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arXiv:1612.03427 · Physical Review A

Monogamy inequality for entanglement and local contextuality

S. Camalet

Published Jun 23, 2017• 14 citations
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Abstract

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.

DOI
10.1103/physreva.95.062329

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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 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 show that the no-disturbance principle imposes a tradeoff between locally contextual correlations violating the Klyachko-Can-Biniciogˇlu-Shumovski inequality and spatially separated correlations violating the Clauser-Horne-Shimony-Holt inequality. The violation of one inequality forbids the violation of the other. We also obtain the corresponding monogamy relation imposed by quantum theory for a qutrit-qubit system. Our results show the existence of fundamental monogamy relations between contextuality and nonlocality that suggest that entanglement might be a particular form of a more fundamental resource.
A unified view of the phenomenon of monogamy exhibited by Bell inequalities and noncontextuality inequalities arising from the no-signaling and no-disturbance principles is presented using the graph-theoretic method introduced by Ramanathan et al. [Phys. Rev. Lett. 109, 050404 (2012)]. We propose a hitherto unexplored tradeoff relation, namely, Bell inequalities that do not exhibit monogamy features of their own can be activated to be monogamous by the addition of a local contextuality term. This is illustrated by means of the well-known I3322{\mathcal{I}}_{3322}I3322​ inequality and reveals a resource trade-off between bipartite correlations and the local purity of a single system. In the derivation of no-signaling monogamies, we uncover a unique feature, namely, that two-party Bell expressions that are trivially classically saturated can become nontrivial upon the addition of an expression involving a third party with a single measurement input.
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.
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