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arXiv:1303.0128 · Physical Review A

True multipartite entanglement Hardy test

Ramij Rahaman, Marcin Wieśniak, Marek Żukowski

Published Dec 24, 2014• 13 citations
arXivHTMLPDF

Abstract

Quantum mechanics allows systems to be entangled with each other, which results in stronger than classical correlations. Many methods of identifying entanglement have been proposed over years, most of which are based on violating some statistical inequalities. In this work we extend the idea formed by Hardy, in which entanglement is not identified with the use of statistical inequalities, but by simultaneous satisfaction of certain conditions. We show that this variant of the Hardy paradox relying on marginal probabilities can be resolved only by true multipartite entangled states. Also, the state resolving this paradox for given local measurements is pure and unique in the case of qubit collections.

DOI
10.1103/physreva.90.062338

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Quantum mechanics allows systems to be entangled with each other, which results in stronger than classical correlations. Many methods of identifying entanglement have been proposed over years, most of which are based on violating some statistical inequalities. In this work we extend the idea formed by Hardy, in which entanglement is not identified with the use of statistical inequalities, but by simultaneous satisfaction of certain conditions. We show that this variant of the Hardy paradox relying on marginal probabilities can be resolved only by true multipartite entangled states. Also, the state resolving this paradox for given local measurements is pure and unique in the case of qubit collections.
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.
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.
Nonlocal correlations as revealed by the violations to Bell inequalities are incompatible with local models without any nonlocal correlations. However some tripartite entangled states, e.g., symmetric pure states, exhibit a stronger nonlocality even incompatible with hybrid local/nonlocal models allowing nonlocal correlations between any two parties. Here we propose a three-particle Hardy-type test without inequality, as illustrated by a gedanken experiment, that is failed by all non-signaling local models while is passed by all tripartite entangled asymmetric pure states. Our result implies that every tripartite entangled state, regardless of the dimensions of the underlying Hilbert spaces, passes a Hardy-type test and therefore is genuine tripartite nonlocal. Maximal success probability and a generalization to multipartite cases of our test are also presented.
Abstract The quantum entanglement as one of the important resources has been verified by using different local models. There is no efficient method to verify single multipartite entanglement that is not generated by multisource quantum networks with local operations and shared randomness. The goal in this work is to solve this problem. First, a new local model is proposed for describing all the states that can be generated by using distributed entangled states and shared randomness without classical communication. This model is stronger than the biseparable model and implies new genuinely multipartite entanglement. With the present local model it is proved that all the permutationally symmetric entangled pure states are new genuinely multipartite entanglement. It is further proved that the new feature holds for all the multipartite entangled pure states in the biseparable model with the dimensions of local systems being not larger than three. The new multipartite entanglement is also robust against general noises. Finally, a simple Bell inequality is provided to verify new genuinely multipartite entangled pure qubit states in the present model. The results show new insight into featuring the genuinely multipartite entanglement in the distributive scenarios.
Here, we present the most general framework for n-particle Hardy's paradoxes, which include Hardy's original one and Cereceda's extension as special cases. Remarkably, for any n≥3, we demonstrate that there always exist generalized paradoxes (with the success probability as high as 1/2^{n-1}) that are stronger than the previous ones in showing the conflict of quantum mechanics with local realism. An experimental proposal to observe the stronger paradox is also presented for the case of three qubits. Furthermore, from these paradoxes we can construct the most general Hardy's inequalities, which enable us to detect Bell's nonlocality for more quantum states.
The correlations between local measurement results are characteristic for multi-partite quantum systems. Under the condition of local realism, which assumes that properties of physical systems exist independent of being measured, and that space-like separated actions on the system cannot influence each other, these correlations are constrained by Bell-inequalities. Quantum mechanics predicts correlations which are not explainable under these two assumptions. The resulting violation of Bell inequalities was originally studied for pairs of entangled particles (Einstein et al.,1935). Meanwhile these ideas were extended to three-and more partite quantum states in particular with respect to the different kinds of multi-particle entanglement and their implications on the obtainable correlations (Mermin, 1990). Here we present a method to construct characteristic Bell inequalities for particular four-qubit states. These inequalities are a versatile tool, not only to provide evidence for non-locality, but also for the characterization of the states, with respect to detection of genuine multipartite entanglement and to state discrimination. Exemplarily we apply our approach to the symmetric four qubit Dicke-state (Dicke, 1954) with two excitations. (A symmetric N-qubit Dicke state with M excitations is the equal superposition of N-qubit pure product states having M logical 1 's and (N-M) logical 0's.) Experimentally the state was observed with photons as carrier of the qubits, using polarization encoding.