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AIP conference proceedings

Average iterations of accessible nonlinear witnesses

Juan Miguel Arrazola, Oleg Gittsovich, Norbert Lütkenhaus

Published Jan 1, 2014• 0 citations
Publisher

Abstract

We present a construction of accessible nonlinear witnesses in terms of an averaged iteration. These witnesses will always detect more states than their linear counterparts. We also review sufficient conditions so that their values can be computed from the same experimental data as their linear counterparts. Finally, we prove that these nonlinear witnesses detect an extensive set of states whose entanglement is detected by the partial transposition criteria.

DOI
10.1063/1.4903118

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We present a construction of accessible nonlinear witnesses in terms of an averaged iteration. These witnesses will always detect more states than their linear counterparts. We also review sufficient conditions so that their values can be computed from the same experimental data as their linear counterparts. Finally, we prove that these nonlinear witnesses detect an extensive set of states whose entanglement is detected by the partial transposition criteria.
Verification of entanglement is an important tool to characterize sources and devices for use in quantum computing and communication applications. In a vast majority of experiments entanglement witnesses (EWs) are used in order to prove the presence of entanglement in a quantum state. EWs can be constructed from available measurement results and do not require a reconstruction of the whole density matrix (full tomography), which is especially valuable for high-dimensional systems. We provide a method to construct accessible nonlinear EWs, which incorporate two important properties. First, they improve on linear EWs in the sense that each nonlinear EW detects more entangled states than its linear counterpart and therefore allows the verification of entanglement without critical dependence on having found the ``right'' linear witness. Second, they can be evaluated using exactly the same data as for the evaluation of the original linear witness. This allows a reanalysis of published experimental data to strengthen statements about entanglement verification without the requirement to perform additional measurements. These particular properties make the accessible nonlinear EWs attractive for the implementations in current experiments, for they can also enhance the statistical significance of the entanglement verification.
Entanglement witnesses (EWs) are fundamental tools for detecting entanglement. However, traditional linear witnesses often fail to identify most of the entangled states. In this work, we construct a family of nonlinear entanglement witnesses by taking the envelope of linear witnesses defined over continuous families of pure bipartite states with fixed Schmidt bases. This procedure effectively ``bends'' the hyperplanes associated with linear witnesses into curved hypersurfaces, thereby extending the region of detectable entangled states. The resulting conditions can be expressed in terms of the positive semidefiniteness of a family of matrices, whose principal minors define a hierarchy of increasingly sensitive detection criteria. We show that this construction is not limited to the transposition map and generalizes naturally to arbitrary positive but not completely positive maps, leading to nonlinear analogs of general entanglement witnesses. We emphasize that the required measurements remain experimentally accessible, as the nonlinear criteria are still formulated in terms of expectation values over local operator bases. Through both analytical and numerical examples, we demonstrate that the proposed nonlinear witnesses outperform their linear counterparts in detecting entangled states which may evade individual linear EWs in the construction. This approach offers a practical and conceptually elegant enhancement to entanglement detection in finite-dimensional systems.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
Abstract Entanglement witnesses (EWs) are one of the most effective methods to detect entanglement. It is known that nonlinear EW provide better entanglement detection than their linear counterparts, in that the former detect a strictly larger subset of entangled states than the latter. Whether linear or nonlinear, the method is measurement-device dependent, so that imperfect measurements may cause false certification of entanglement in a shared state. Measurement-device-independent EW provide an escape from such measurement dependence of the entanglement detection for linear EW. Here we present measurement-device-independent nonlinear EW for non-positive partial transpose entangled states as well as for bound entangled states with positive partial transpose. Although the witness considered herein does not detect a larger set of entangled states than other nonlinear EW, it is more efficient in that it never leads to a false detection corresponding to wrong measurements. The constructed measurement-device-independent nonlinear EW certify the entanglement of the same sets of entangled states as their device-dependent parents do, and therefore are better than the linear EW, device-independent or otherwise.
Nonlinear entanglement witnesses constructed from multiple linear entanglement witnesses and multiple copies of quantum states have recently been proposed as a powerful tool for entanglement detection. In this work, we show, via an explicit counterexample, that the fineness of linear witnesses generally fails to transfer to their tensor-product nonlinear counterparts. For the canonical family of nonlinear witnesses in the form of (αI-L)⊗(βI-T), we rigorously prove that the optimal nonlinear witness is uniquely attained with weakly optimal parameters of α=λmax(L) and β=λmax(T). Meanwhile, we analytically demonstrate that the self-tensor products of two representative linear witnesses fail to detect any entangled state. The question of whether a nonlinear entanglement witness capable of detecting entanglement can be constructed by tensoring a linear witness with itself remains open.