Nonlocality and entanglement in qubit systems

Abstract

Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, {\it Gisin's Theorem}), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio $R=1/Tr({\rho }^2)$ or their maximum eigenvalue ${\lambda }_{max}$. This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four qubit systems, where nonlocality for generalized Greenberger-Horne-Zeilinger states of arbitrary number of parties is computed.

16 pages, 3 figures

References in corpus (8)

• Device-independent security of quantum cryptography against collective attacks
• Device-independent quantum key distribution secure against collective attacks
• Testing the Hilbert space dimension
• Grothendieck's constant and local models for noisy entangled quantum states
• More efficient Bell inequalities for Werner states
• Nonlocality and entanglement in the XY model
• Maximally Entangled Mixed States and Conditional Entropies
• Teleportation capability, distillability, and nonlocality on three-qubit states

Cited by in corpus (17)

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• Entanglement as upper bounded for the nonlocality of a general two-qubit system
• Complementarity between Tripartite Quantum Correlation and Bipartite Bell Inequality Violation in Three Qubit States
• Bell violation versus geometric measure of quantum discord and their dynamical behavior
• Bell-CHSH function approach to quantum phase transitions in matrix product systems
• Intrinsic degree of coherence of two-qubit states and measures of two-particle quantum correlations
• Attainable and usable coherence in X states over Markovian and non-Markovian channels
• Quantifying necessary quantum resources for nonlocality
• Quantifying Bell nonlocality of a pure two-qudit state via its entanglement
• On the Activation of Quantum Nonlocality
• Bound entangled Bell diagonal states of unequal local dimensions, and their witnesses
• Entanglement versus Bell nonlocality of quantum nonequilibrium steady states
• Maximal Clauser-Horne-Shimony-Holt violation for qubit-qudit states
• Bell Non-Locality in Many Body Quantum Systems with Exponential Decay of Correlations
• Inequality relation between entanglement and Bell nonlocality for arbitrary two-qubit states
• Comparing coherence measures for X states: Can quantum states be ordered based on quantum coherence?