In contrast to the wide-spread opinion that any separable quantum state satisfies every classical probabilistic constraint, we present a simple example where a separable quantum state does not satisfy the original Bell inequality although the latter inequality, in its perfect correlation form, is valid for all joint classical measurements. In a very general setting, we discuss inequalities for joint experiments upon a bipartite quantum system in a separable state. We derive quantum analogues of the original Bell inequality and specify the conditions sufficient for a separable state to satisfy the original Bell inequality. We introduce the extended CHSH inequality and prove that, for any separable quantum state, this inequality holds for a variety of linear combinations.
Using the Hilbert-Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EW ) for n (>2) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we gi…
I derive separability inequalities for Bell correlations of observables in arbitrary pure or mixed $N$ Qudit states in $D^N$-dimensional state space. I find states (a continuum of states if $D>3$) including maximally entangled states which violate these inequalities by a factor $2^{N-1}$ ; local reality Bell inequalities are much weaker, their maximum violation being by a factor $2^{(N-1)/2}$. The separability inequalities allow tests of entanglement of unknown states using only the measured correlations .
Last years, bounds on the maximal quantum violation of general Bell inequalities were intensively discussed in the literature via different mathematical tools. In the present paper, we analyze quantum violation of general Bell inequalities via the LqHV (local quasi hidden variable) modelling framework, correctly reproducing the probabilistic description of every quantum correlation scenario. The LqHV mathematical framework allows us to derive for all d and N a new upper bound (2d-1)^{N-1} on the maximal violation by an N-qudit state of all general Bell inequalities, also, new upper bounds on t…
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the com…
With a probability of success of $95 \%$ we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be sepa…
Noncommuting observables cannot be simultaneously measured, however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fréchet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of n…
Entanglement of quantum states is absolutely essential for modern quantum sciences and technologies. It is natural to extend the notion of entanglement to quantum observables dual to quantum states. For quantum states, various separability criteria have been proposed to determine whether a given state is entangled. In this Letter, we propose a separability criterion for specific quantum effects (binary observables) that can be regarded as a dual version of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for quantum states. The violation of the dual version of the Bell-CHSH inequalit…
Beyond the standard entropic inequalities: stronger scalar separability criteria and their applications
最近，它表明，如果一个给定的状态满足约化准则，它也必须满足已知的熵不等式。现在的问题是，假设满足基于正映射但不完全正映射的更强准则，是否有可能导出一些比熵不等式更强的标量不等式。在本文中，我们证明了在某些假设下，扩展的约化准则[H.- P. Breuer，Phys. Rev. Lett 97，080501（2006）; W. Hall，J.Phys.A40，6183（2007）]导致了一些类似熵的不等式，这些不等式比它们的熵对应物强得多。与其他可分离性判据的比较表明，这种方法可能会导致强标量判据检测可分离纠缠和束缚纠缠。特别地，在SO（3）不变态的情况下，它表明，目前的不等式检测纠缠的区域中的纠缠证人的基础上扩展的约简映射失败。还应该强调的是，在$2\otimes N$状态的情况下，导出的不等式有效地检测纠缠，而扩展的约简映射在作用于量子比特子系统时是无用的。此外，有一个自然的方式来构建一个多副本纠缠证人的基础上导出的不等式，所以，在原则上，有实验实现的可能性。一些开放的问题和进一步研究的可能性进行了概述。
Finding all Bell inequalities for a given number of parties, measurement settings, and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins-Gisin-Linden-…
Quantum nonlocality, one of the most important features of quantum mechanics, is normally connected in experiments with the violation of Bell-Clauser-Horne (Bell-CH) inequalities. We propose effective methods for the rearrangement and linear inequality to prove a large variety of Bell-CH inequalities. We also derive a set of Bell-CH inequalities by using these methods which can be violated in some quantum entangled states.
