Necessary and sufficient conditions for bipartite entanglement

Sperling, Jan; Vogel, W.

doi:10.1103/physreva.79.022318

Cited by 91 publications

(111 citation statements)

References 21 publications

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“…[3] and [4]. Entanglement of general states can be verified by involved generalizations of the PT tests [14], or, based on entanglement witnesses [15], by optimized entanglement conditions [16]. However, it has been proven that for any entangled state there exist subspaces of finite dimension in which the entanglement already exists [17].…”

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confidence: 99%

Entangled qubits in a non-Gaussian quantum state

et al. 2011

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We experimentally generate and tomographically characterize a mixed, genuinely non-Gaussian bipartite continuous-variable entangled state. By testing entanglement in 2×2-dimensional two-qubit subspaces, entangled qubits are localized within the density matrix, which, first, proves the distillability of the state and, second, is useful to estimate the efficiency and test the applicability of distillation protocols. In our example, the entangled qubits are arranged in the density matrix in an asymmetric way, i.e., entanglement is found between diverse qubits composed of different photon number states, although the entangled state is symmetric under exchanging the modes.

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Section: Introductionmentioning

confidence: 99%

Entangled qubits in a non-Gaussian quantum state

et al. 2011

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“…Hence, there exists an optimal point ja; bi for which ha; bjŴja; bi ¼ 0 [7,14]. The resulting EW is said to be the finest witness in the sense that any further shift of its corresponding hyperplane will lead to an operator whose expectation value becomes negative for some separable states, thus, violating the proper witnessing conditions [12,13]. It is, however, possible to significantly increase the detection power of any test operator by taking into account additional constraints and information about the states under investigation, which effectively reduces the size of the set of viable separable states, see Fig.…”

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“…whereÎ is the identity operator, and g s ¼ supfTrLσ ∶ σ ∈ S sep g. Indeed, it is sufficient to optimize only over pure product states ja; bi [13]. One can also employ a similar recipe using the infimum value g i ¼ inffTrLσ ∶σ ∈ S sep g. This optimization procedure can be geometrically understood as translating the hyperplane corresponding to the test operator until it is tangent to the set of separable states, see Fig.…”

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“…First, adding nonlinear terms to the original witness operator [10]; second, using collective measurements of EWs on multiple copies of the quantum state [11]; and third, optimizing a given witness to tighten the bound on the statistics of separable states as much as possible [12,13]. The latter, which we refer to as standard entanglement witnessing (SEW), is the most common procedure and can be used as a complementary procedure to the first two techniques.…”

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“…Powerful EWs are most commonly constructed by optimizing a Hermitian (and possibly completely positive) test operatorL over the set of separable states as [12,13] …”

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Ultrafine Entanglement Witnessing

et al. 2017

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Entanglement witnesses are invaluable for efficient quantum entanglement certification without the need for expensive quantum state tomography. Yet, standard entanglement witnessing requires multiple measurements and its bounds can be elusive as a result of experimental imperfections. Here, we introduce and demonstrate a novel procedure for entanglement detection which simply and seamlessly improves any standard witnessing procedure by using additional available information to tighten the witnessing bounds. Moreover, by relaxing the requirements on the witness operators, our method removes the general need for the difficult task of witness decomposition into local observables. We experimentally demonstrate entanglement detection with our approach using a separable test operator and a simple fixed measurement device for each agent. Finally, we show that the method can be generalized to higher-dimensional and multipartite cases with a complexity that scales linearly with the number of parties. DOI: 10.1103/PhysRevLett.118.110502 Quantum entanglement provides many advantages beyond classical limits, including quantum communication, computation, and information processing [1,2]. Yet, determining whether a given quantum state is entangled or not is a theoretically and experimentally challenging task [3,4]. In particular, the ideal approach of reconstructing the full quantum state via quantum tomography is practically infeasible for all but the smallest systems.An elegant solution to this problem, known as entanglement witnessing, relies on the geometry of the set of nonentangled (separable) quantum states [2,[5][6][7]. Since these states form a convex set, it is always possible to find a hyperplane such that a given entangled state lies on one side of the hyperplane, while all separable states are on the other side, see Fig. 1. This hyperplane is a so-called entanglement witness (EW) and corresponds to a joint observable that has a bounded expectation value over all separable quantum states. Any quantum state that produces a value beyond the bound must be entangled. This simplification, however, comes at a cost: first, different entangled states in general require different EWs to be detected; second, not every EW can be practically realized, i.e., can be decomposed into operators corresponding to available local measurement devices (See also Refs. [7][8][9] for examples of the reverse procedure: constructing EWs from local observables); third, when such a decomposition is possible, it might require multiple measurement devices (with multiple settings) to be implemented; and fourth, witnessing bounds can be elusive in the presence of experimental imperfections. Consequently, the goal is to construct EWs that have a simple decomposition and, at the same time, detect a large set of entangled states.There are three main techniques to improve EWs. First, adding nonlinear terms to the original witness operator [10]; second, using collective measurements of EWs on multiple copies of the quantum state [11]; and third, op...

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The Weirdness Theorem and the Origin of Quantum Paradoxes

2021

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We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics.

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Necessary and sufficient conditions for bipartite entanglement

Cited by 91 publications

References 21 publications

Entangled qubits in a non-Gaussian quantum state

Entangled qubits in a non-Gaussian quantum state

Ultrafine Entanglement Witnessing

The Weirdness Theorem and the Origin of Quantum Paradoxes

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