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Physical Review A

Shareability of correlations in multiqubit states: Optimization of nonlocal monogamy inequalities

Josep Amengual i Batle, Mosayeb Naseri, MAHMOOD GHORANNEVISS, Ahmed Farouk, Majid Alkhambashi, Mohamed Elhoseny

Published Mar 20, 2017• 46 citations
Publisher

Abstract

It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.

DOI
10.1103/physreva.95.032123

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It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.
In multiparty quantum systems, the monogamy inequality proposes an upper bound on the distribution of bipartite quantum correlation between a single party and each of the remaining parties in the system, in terms of the amount of quantum correlation shared by that party with the rest of the system taken as a whole. However, it is well known that not all quantum correlation measures universally satisfy the monogamy inequality. In this work, we aim at determining the nontrivial value by which the monogamy inequality can be violated by a quantum correlation measure. Using an information-theoretic complementarity relation between the normalized purity and quantum correlation in any given multiparty state, we obtain a nontrivial lower bound on the negative monogamy score for the quantum correlation measure. In particular, for the three-qubit states the lower bound is equal to the negative von Neumann entropy of the single qubit reduced density matrix. We analytically examine the tightness of the derived lower bound for certain nnn-qubit quantum states. Further, we report numerical results of the same for monogamy violating correlation measures using Haar uniformly generated three-qubit states.
Entanglement and Bell nonlocality have different sharing constraints across subsystems of a multipartite quantum system. We examine simultaneous violations of Bell inequalities with two-outcome observables and two or three measurement settings per party. We identify configurations whose simultaneous violations can be demonstrated on IBM Quantum hardware, and derive bounds and trade-off relations for several polygamous configurations. The results connect experimental capabilities of quantum computers with mathematical methods for studying the limits of quantum correlations.
Two overlapping bipartite binary Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signaling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations of a few inequalities at once. Here we systematically study monogamy relations between correlation Bell inequalities both within quantum theory and under the sole assumption of no signaling. We first investigate the trade-offs in Bell violations arising from the uncertainty relation for complementary binary observables, and exhibit several network configurations in which a tight trade-off arises in this fashion. We then derive a tight trade-off relation which cannot be obtained from the uncertainty relation showing that it does not capture monogamy entirely. The results are extended to Bell inequalities involving different numbers of parties and find applications in device-independent secret sharing and device-independent randomness extraction. Although two multipartite Bell inequalities may be violated simultaneously, we show that genuine multiparty nonlocality, as evidenced by a generalized Svetlichny inequality, does exhibit monogamy property. Finally, using the relations derived we reveal the existence of flat regions in the set of quantum correlations.
The entangled multipartite systems, specially in pure states, exhibit the phenomenon of entanglement monogamy. Such systems also display the phenomenon of Bell nonlocality. Like entanglement monogamy relations, there are Bell monogamy relations. These relations suggest a sharing of nonlocality across the subsystems. The nonlocality, as characterized by Bell inequalities, of one subsystem limits the nonlocality exhibited by another subsystem. We show that the Bell monogamy relations can be violated by using local filtering operations. We consider permutation-symmetric multipartite pure states, in particular WWW states, to demonstrate the violation.
Monogamy is a nonclassical property that restricts the sharability of quantum correlation among the constituents of a multipartite quantum system. Quantum correlations may satisfy or violate monogamy for quantum states, which was tested mainly for three-qubit states. Here we establish a sufficient condition for monogamy of arbitrary quantum correlation measures of states of an arbitrary number of parties, using which and further numerical results, we obtain evidence for monogamy of measures such as distillable entanglement and relative entropy of entanglement, which are physically important but mathematically intractable, for almost all quantum states of a moderate number of parties. The result is generic and holds for a large class of quantum correlation measures. Nonetheless, we identify important zero Haar measure classes of pure states that remain nonmonogamous with respect to quantum discord and quantum work deficit, irrespective of the number of qubits.
Quantum mechanics imposes limits on the statistics of certain observables. Perhaps the most famous example is the uncertainty principle. Similar trade-offs also exist for the simultaneous violation of multiple Bell inequalities. In the simplest case of three observers, it has been shown that if two observers violate a Bell inequality, then none of them can violate any Bell inequality with the third observer, a property called monogamy of Bell violations. Forms of Bell monogamy have been linked to the no-signaling principle, and the inability of simultaneous violations of all inequalities is regarded as their fundamental property. Here, we show that the Bell monogamy does not hold universally and that in fact the only monogamous situation exists for only three observers. Consequently, the nature of quantum nonlocality is truly polygamous. We present a systematic methodology for identifying quantum states, measurements, and tight Bell inequalities that do not obey the monogamy principle for any number of more than three observers. The identified polygamous inequalities enable any subset of [Formula: see text] observers to reveal nonlocality, which is also shown experimentally by measuring Bell-type correlations of six-photon Dicke states. Our findings may be exploited for multiparty quantum key distribution as well as simultaneous self-testing of multiple nodes in quantum networks.