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arXiv:1901.07946 · Physical Review A

Entanglement detection with scrambled data

Timo Simnacher, Nikolai Wyderka, René Schwonnek, Otfried Gühne

Published Jun 27, 2019• 1 citation
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Abstract

In the usual entanglement detection scenario the possible measurements and the corresponding data are assumed to be fully characterized. We consider the situation where the measurements are known, but the data is scrambled, meaning the assignment of the probabilities to the measurement outcomes is unknown. We investigate in detail the two-qubit scenario with local measurements in two mutually unbiased bases. First, we discuss the use of entropies to detect entanglement from scrambled data, showing that Tsallis and R\'enyi entropies can detect entanglement in our scenario, while the Shannon entropy cannot. Then, we introduce and discuss scrambling-invariant families of entanglement witnesses. Finally, we show that the set of nondetectable states in our scenario is nonconvex and therefore in general hard to characterize.

DOI
10.1103/physreva.99.062339

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In the usual entanglement detection scenario the possible measurements and the corresponding data are assumed to be fully characterized. We consider the situation where the measurements are known, but the data is scrambled, meaning the assignment of the probabilities to the measurement outcomes is unknown. We investigate in detail the two-qubit scenario with local measurements in two mutually unbiased bases. First, we discuss the use of entropies to detect entanglement from scrambled data, showing that Tsallis and R'enyi entropies can detect entanglement in our scenario, while the Shannon entropy cannot. Then, we introduce and discuss scrambling-invariant families of entanglement witnesses. Finally, we show that the set of nondetectable states in our scenario is nonconvex and therefore in general hard to characterize.
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.
We experimentally demonstrate a method for detection of entanglement via construction of entanglement witnesses from a limited fixed set of local measurements ( M ). Such a method does not require a priori knowledge about the form of the entanglement witnesses. It is suitable for a scenario where a full state tomography is not available, but the only resource is a limited set of M . We demonstrate the method on pure two-qubit entangled states and mixed two-qubit entangled states, which emerge from photonic implementation of controllable quantum noisy channels. The states we select are motivated by realistic experimental conditions, and we confirm it works well for both cases. Furthermore, possible generalizations to higher-dimensional bipartite systems have been considered, which can potentially detect both decomposable and indecomposable entanglement witnesses. Our experimental results show perfect validity of the method, which indicates that even a limited set of local measurements can be used for quick entanglement detection and further provide a practical test bed for experiments with entanglement witnesses.
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We investigate correlations among complementary observables. In particular, we show how to take advantage of mutually unbiased bases for the efficient detection of entanglement in arbitrarily high-dimensional, multipartite, and continuous-variable quantum systems. The introduced entanglement criteria are relatively easy to implement experimentally since they require only a few local measurement settings. In addition, we establish a link between the separability problem and the maximum number of mutually unbiased bases---opening an additional avenue in this long-standing open problem.
We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion for bipartite systems with equal dimensions is based on measurements from subsets of a quantum 2-design, e.g. mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for quantum state tomography. Secondly, our method is twice as effective as standard witnesses in the sense that both upper and lower bounds can be derived. Thirdly, the scheme can be readily applied to measurement-device-independent scenarios.
We present the generalized state-dependent entropic uncertainty relations in a multiple measurements setting and the optimal lower bound is obtained by considering different measurement sequences. We then apply this uncertainty relation to witness entanglement and give the experimentally accessible lower bounds on both bipartite and tripartite entanglements. This method of detecting entanglement is applied to a physical system of two particles on a one-dimensional lattice and Greenberger-Horne-Zeilinger (GHZ)-Werner state. It is shown that, for measurements that are not in mutually unbiased bases, this generalized entropic uncertainty relation is superior to the previous state-independent one in entanglement detection. Furthermore, we conduct a demonstration of multipartite entanglement detection of GHZ states up to ten qubits on the Quafu cloud quantum computation platform. Our results might play important roles in detecting multipartite entanglement experimentally.