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arXiv:1110.3026 · Physical Review A

Monogamy of quantum correlations in three-qubit pure states

Sudha, A. R. Usha Devi, A. K. Rajagopal

Published Jan 10, 2012• 44 citations
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Abstract

The limitation on the shareability of quantum entanglement over several parties, the so-called monogamy of entanglement, is an issue that has received considerable attention from the quantum information community over the last decade. A natural question of interest in this connection is whether monogamy of correlations is true for correlations other than entanglement. This issue is examined here by choosing quantum deficit, proposed by A. K. Rajagopal and R. W. Rendell [Phys. Rev. A 66, 022104 (2002)], an operational measure of correlations. In addition to establishing the polygamous nature of the class of three-qubit symmetric pure states characterized by two distinct Majorana spinors (to which the $W$ states belong), those with three distinct Majorana spinors [to which Greenberger-Horne-Zeilinger (GHZ) states belong] are shown to either obey or violate monogamy relations. While the generalized $W$ states can be monogamous or polygamous, the generalized GHZ states exhibit monogamy with respect to quantum deficit. The issue of using monogamy conditions based on quantum deficit to witness the states belonging to stochastic local operations and classical communication (SLOCC) inequivalent classes is discussed in light of these results.

DOI
10.1103/physreva.85.012103

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The limitation on the shareability of quantum entanglement over several parties, the so-called monogamy of entanglement, is an issue that has received considerable attention from the quantum information community over the last decade. A natural question of interest in this connection is whether monogamy of correlations is true for correlations other than entanglement. This issue is examined here by choosing quantum deficit, proposed by A. K. Rajagopal and R. W. Rendell [Phys. Rev. A 66, 022104 (2002)], an operational measure of correlations. In addition to establishing the polygamous nature of the class of three-qubit symmetric pure states characterized by two distinct Majorana spinors (to which the WWW states belong), those with three distinct Majorana spinors [to which Greenberger-Horne-Zeilinger (GHZ) states belong] are shown to either obey or violate monogamy relations. While the generalized WWW states can be monogamous or polygamous, the generalized GHZ states exhibit monogamy with respect to quantum deficit. The issue of using monogamy conditions based on quantum deficit to witness the states belonging to stochastic local operations and classical communication (SLOCC) inequivalent classes is discussed in light of these results.
Quantum correlations are expected to respect all the conditions required for them to be good measures of quantumness in the bipartite scenario. In a multipartite setting, sharing entanglement between several parties is restricted by the monogamy of entanglement. We take over the concept of monogamy to an information-theoretic quantum correlation measure, and find that it violates monogamy in general. Using the notion of interaction information, we identify necessary and sufficient conditions for the measure to obey monogamy, for arbitrary pure and mixed quantum states. We show that while three-qubit generalized Greenberger-Horne-Zeilinger states follow monogamy, generalized WWW states do not.
We examine here the proposition that all multiparty quantum states can be made monogamous by considering positive integral powers of any quantum correlation measure. With Rajagopal-Rendell quantum deficit as the measure of quantum correlations for symmetric 3-qubit pure states, we illustrate that monogamy inequality is satisfied for higher powers of quantum deficit. We discuss the drawbacks of this inequality in quantification of correlations in the state. We also prove a monogamy inequality in higher powers of classical mutual information and bring out the fact that such inequality needs not necessarily imply restricted shareability of correlations. We thus disprove the utility of higher powers of any correlation measure in establishing monogamous nature in multiparty quantum states.
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.
In contrast with entanglement, as measured by concurrence, in general, quantum discord does not possess the property of monogamy; that is, there is no tradeoff between the quantum discord shared by a pair of subsystems and the quantum discord that both of them can share with a third party. Here, we show that, as far as monogamy is considered, quantum discord of pure states is equivalent to the entanglement of formation. This result allows one to analytically prove that none of the pure three-qubit states belonging to the subclass of WWW states is monogamous. A suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given. Finally, we prove that, for rank 2 two-qubit states, discord and classical correlations are bounded from above by single-qubit von Neumann entropies.
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.
The multi-qubit GHZ state possesses tangles with elegant transformation properties under stochastic local operations and classical communication. Since almost all pure 3-qubit states are connected to the GHZ state via SLOCC, we derive a necessary and sufficient achievability inequality on arbitrary 3-qubit tangles, which is a strictly stronger constraint than both the monogamy inequality and the marginal eigenvalue inequality. We then show that entanglement shared with any single party in the n-qubit GHZ SLOCC equivalence class is precisely accounted for by the sum of its k-tangles, recently coined the strong monogamy equality, acknowledging competing but agreeing definitions of the k-tangle on this class, one of which is then computable for arbitrary mixed states. Strong monogamy is known to not hold arbitrarily, and so we introduce a unifying outlook on entanglement constraints in light of basic real algebraic geometry.