Hardy’s non-locality provides a proof of the incompatibility between quantum mechanics and local realism without using Bell inequalities. While this argument has been extensively studied for two- and three-qubit systems, a detailed analysis of the four-qubit case is still lacking. In this work, we investigate Hardy’s non-locality for a four-qubit system within the standard two-setting framework. We explicitly construct the entangled state satisfying the Hardy conditions and determine the measurement settings that maximize the success probability. • We present an explicit analytical construction of a four-qubit Hardy state together with the corresponding projective measurement settings. • The maximum Hardy success probability for four qubits is found to be 0.143, exceeding the values obtained in the two- and three-qubit cases. • The resulting state is non-maximally entangled and exhibits genuine multipartite quantum correlations.
Hardy’s non-locality provides a proof of the incompatibility between quantum mechanics and local realism without using Bell inequalities. While this argument has been extensively studied for two- and three-qubit systems, a detailed analysis of the four-qubit case is still lacking. In this work, we investigate Hardy’s non-locality for a four-qubit system within the standard two-setting framework. We explicitly construct the entangled state satisfying the Hardy conditions and determine the measurement settings that maximize the success probability. • We present an explicit analytical construction of a four-qubit Hardy state together with the corresponding projective measurement settings. • The maximum Hardy success probability for four qubits is found to be 0.143, exceeding the values obtained in the two- and three-qubit cases. • The resulting state is non-maximally entangled and exhibits genuine multipartite quantum correlations.
In this paper the failure of Hardy's nonlocality proof for the class of maximally entangled states is considered. A detailed analysis shows that the incompatibility of the Hardy equations for this class of states physically originates from the fact that the existence of quantum perfect correlations for the three pairs of two-valued observables (D_11,D_21), (D_11,D_22) and (D_12,D_21) [in the sense of having with certainty equal (different) readings for a joint measurement of any one of the pairs (D_11,D_21), (D_11,D_22), and (D_12,D_21)], necessarily entails perfect correlation for the pair of observables (D_12,D_22) [in the sense of having with certainty equal (different) readings for a joint measurement of the pair (D_12,D_22)]. Indeed, the set of these four perfect correlations is found to satisfy the CHSH inequality, and then no violations of local realism will arise for the maximally entangled state as far as the four observables D_ij, i,j = 1,2, are concerned. The connection between this fact and the impossibility for the quantum mechanical predictions to give the maximum possible theoretical violation of the CHSH inequality is pointed out. Moreover, it is generally proved that the fulfillment of all the Hardy nonlocality conditions necessarily entails a violation of the resulting CHSH inequality. The largest violation of this latter inequality is determined.
Nonlocality without inequality is an elegant argument introduced by Hardy for two qubit systems, and later generalised to n qubits, to establish contradiction of quantum theory with local realism. Interestingly, for n = 2 this argument is actually a corollary of Bell-type inequalities, viz., the CH–Hardy inequality involving Bell correlations, but for n greater than two it involves n-particle probabilities more general than Bell-correlations. In this paper, we first derive a chain of completely new local realistic inequalities involving joint probabilities for n qubits and then associated with each such inequality, we provide a new Hardy-type local reality constraint without inequalities. Quantum mechanical maximal violations of the chain of inequalities and of the associated constraints are also studied by deriving appropriate Cirel'son-type theorems. These results involving joint probabilities more general than Bell correlations are expected to provide a new systematic tool to investigate entanglement.
Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.
We test Hardy's paradox of non-locality experimentally on the IBM five-qubit quantum computer for the first time. The quantum circuit is constructed on superconducting qubits corresponding to the original Hardy's test of non-locality. Results confirmed the theory that any non-maximally entangled state of two qubits violates Hardy's Equations, whereas any maximally entangled state and product state of two qubits do not exhibit Hardy's non-locality. We also point out the difficulties associated with the practical implementation of any Hardy's paradox based quantum protocol and propose three performance measures for any two qubits of any quantum computer.
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, 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(ρ 2 ) or their maximum eigenvalue λ 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.