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Register a new userCharacterization and properties of weakly optimal entanglement witnesses
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We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of $W^{wopt}=sigma-c_{sigma}^{max} I$, where $c_{sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzic{a}g {it et al}, Phys. Rev. A {bf 88}, 010301(R) (2013)], and we examine their geometric properties.
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Quantum entanglement lies at the heart of quantum mechanical and quantum information processing. Following the question who emph{witnesses} entanglement witnesses, we show entangled states play as the role of super entanglement witnesses. We show separable states play the role of super super entanglement witnesses and witness other observables than entanglement witnesses. We show that there exists a hierarchy structure of witnesses and there exist witnesses everywhere. Furthermore, we show the properties and characterization of entangled states as super entanglement witnesses. By the role of super witnesses of entangled states, we immediately find the question when different entanglement witnesses can detect the same entangled states [{it Phys. Lett. A }{bf 356} 402 (2006)] is the same as the question when different entangled states can be detected by the same entanglement witnesses [{it Phys. Rev. A} {bf 75} 052333 (2007)]. By the role of witnesses, we define finer entangled states and optimal entangled states. The definition gives a nonnumeric measurement of entanglement and an unambiguous discrimination of entangled states, and the procedure of optimization for a general entangled state $rho$ is just finding the best separable approximation (BSA) to $rho$ in [ {it Phys. Rev. Lett.} {bf 80} 2261 (1998)].}
We provide a class of optimal nondecomposable entanglement witnesses for 4N x 4N composite quantum systems or, equivalently, a new construction of nondecomposable positive maps in the algebra of 4N x 4N complex matrices. This construction provides natural generalization of the Robertson map. It is shown that their structural physical approximations give rise to entanglement breaking channels.
Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the inertias of EWs, i.e., the triplet of the numbers of negative, zero, and positive eigenvalues respectively. We focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit-qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X-states are clarified.
The ultrafine entanglement witness, introduced in [F. Shahandeh, M. Ringbauer, J.C. Loredo, and T.C. Ralph, Phys. Rev. Lett. textbf{118}, 110502 (2017)], can seamlessly and easily improve any standard entanglement witness. In this paper, by combining the constraint and the test operators, we rotate the hyperplane determined by the test operator and improve further the original ultrafine entanglement witness. In particular, we present a series of new ultrafine entanglement witnesses, which not only can detect entangled states that the original ultrafine entanglement witnesses cannot detect, but also have the merits that the original ultrafine entanglement witnesses have.
How to detect quantum correlations in bi-partite scenarios using a split many-body system and collective measurements on each party? We address this question by deriving entanglement witnesses using either only first or first and second order moments of local collective spin components. In both cases, we derive optimal witnesses for spatially split spin squeezed states in the presence of local white noise. We then compare the two optimal witnesses with respect to their resistance to various noise sources operating either at the preparation or at the detection level. We finally evaluate the statistics required to estimate the value of these witnesses when measuring a split spin-squeezed Bose-Einstein condensate. Our results can be seen as a step towards Bell tests with many-body systems.
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