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Register a new userThe hierarchies of witnesses and the properties and characterization of entangled states as super entanglement witnesses
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Bang-Hai Wang
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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)].}
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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.
Quantum entanglement lies at the heart of quantum mechanics and quantum information processing. In this work, we show a new framework where entangled states play the role of witnesses. We extend the notion of entanglement witnesses, developing a hierarchy of witnesses for classes of observables. This hierarchy captures the fact that entangled states act as witnesses for detecting entanglement witnesses and separable states act as witnesses for the set of non-block-positive Hermitian operators. Indeed, more hierarchies of witnesses exist. We introduce the concept of emph{finer} and emph{optimal} entangled states. These definitions not only give an unambiguous and non-numeric quantification of entanglement and a new perspective on edge states but also answer the open question of what the remainder of the best separable approximation of a density matrix. Furthermore, we classify all entangled states into disjoint families with optimal entangled states at its heart. This implies that we can focus only on the study of a typical family with optimal entangled states at its core when we investigate entangled states. Our framework also assembles many seemingly different findings with simple arguments that do not require lengthy calculations.
The problem of demonstrating entanglement is central to quantum information processing applications. Resorting to standard entanglement witnesses requires one to perfectly trust the implementation of the measurements to be performed on the entangled state, which may be an unjustified assumption. Inspired by the recent work of F. Buscemi [Phys. Rev. Lett. 108, 200401 (2012)], we introduce the concept of Measurement-Device-Independent Entanglement Witnesses (MDI-EWs), which allow one to demonstrate entanglement of all entangled quantum states with untrusted measurement apparatuses. We show how to systematically obtain such MDI-EWs from standard entanglement witnesses. Our construction leads to MDI-EWs that are loss-tolerant, and can be implemented with current technology.
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.
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