Quantum Entanglement: Bell's Inequality Trivially Violated Also Classically

Biao Wu

Volkan Erol

The violation of Bell's theorem is a very simple way to see that there is no underlying classical interpretation of quantum mechanics. The measurements made on the photons shows that light signal (information) could travel between them, hence completely eliminating any chance that the result was due to anything other than entanglement. Entanglement has been studied extensively for understanding the mysteries of non-classical correlations between quantum systems. It was found that violation of Bell's inequalities could be trivially calculated and for sets of nonmaximally entangled states of two qubits, comparing these entanglement measures may lead to different entanglement orderings of the states. On the other hand, although it is not an entanglement measure and not monotonic under local operations, due to its ability of detecting multipartite entanglement, quantum Fisher information (QFI) has recently received an intense attraction generally with entanglement in the focus. In this work, we visit violation of Bell's inequalities problem with a different approach. Generating a thousand random quantum states and performing an optimization based on local general rotations of each qubit, we calculate the maximal QFI for each state. We analyze the maximized QFI in comparison with violation in Bell's inequalities and we make similar comparison of this violation with commonly studied entanglement measures, negativity and relative entropy of entanglement. We show that there are interesting orderings for system states.

Shiv Akshar Yadavalli, Nikola Andrejić, Ravi Kunjwal

Measurements in quantum theory can fail to be jointly measurable. Like entanglement, this incompatibility of measurements is necessary but not sufficient for violating Bell inequalities. The (in)compatibility relations among a set of measurements can be represented by a joint measurability structure, i.e., a hypergraph whose vertices denote measurements and hyperedges denote all and only compatible sets of measurements. Since incompatibility is necessary for a Bell violation, the joint measurability structure on each wing of a Bell experiment must necessarily be non-trivial, i.e., it must admit a subset of incompatible vertices. Here we show that for any non-trivial joint measurability structure with a finite set of vertices, there exists a quantum realization with a set of measurements that enables a Bell violation, i.e., given that Alice has access to this incompatible set of measurements, there exists a set of measurements for Bob and an entangled state shared between them such that they can jointly violate a Bell inequality. Hence, a non-trivial joint measurability structure is not only necessary for a Bell violation, but also sufficient.

Dong Ding, Ying-Qiu He, Fengli Yan +1 more

Yeong-Cherng Liang

Pereira Ls

For many years it was believed that entanglement and violation of the Bell inequalities were the same peculiar aspect of Quantum Mechanics: non-locality. Nowadays, it is known that this equivalence is not complete, being true only for pure states; in other words, there are entangled mixed states that do not violate the Bell inequalities. Thus it is interesting to analyze properties of states which exhibit such difference, in order to better understand it. Here we present such a study of the connection between entanglement and violation of the Bell inequalities in Quantum Phase Transitions. More pecifically, we analyze this relation for two particles in an infinity one-dimensional spin-12 chain described by the XXZ model. We show that the two particles do not violate the Bell-CHSH inequality for any value of anisotropy, even in regions where they are entangled. Therefore, the entanglement that is relevant in Quantum Phase Transitions is not a trivial one , since it is not revealed by the Bell-CHSH inequality. We also demonstrate that the study of the Bell-CHSH inequality can indicate not only the first order phase transition as the infinity order one, while the fundamental energy per particle and the concurrence can only indicate the first order phase transition.