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arXiv:1803.02708 · New Journal of Physics

Linking entanglement detection and state tomography via quantum 2-designs

Joonwoo Bae, Beatrix C. Hiesmayr, Daniel McNulty

Published Dec 17, 2018• 44 citations
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Abstract

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.

DOI
10.1088/1367-2630/aaf8cf

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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.
In this work, we show that measurement in quantum 2-design, such as mutually unbiased bases (MUBs) or symmetric, informationally complete states (SICs), improves the capability of detecting entangled states both theoretically and experimentally. On the theoretical side, we show that measurement in quantum 2-design can detect entangled states twice compared to entanglement witnesses. On the implementation side, we present the scheme of entanglement detection with two detectors only of a Hong-Ou-Mandel interferometer. The experimental scheme applies single-copy level measurement followed by post-processing of measurement outcomes, which is feasible with current technologies.
Mutually unbiased bases, mutually unbiased measurements and general symmetric informationally complete measurements are three related concepts in quantum information theory. We investigate multipartite systems using these notions and present some criteria detecting entanglement of arbitrary high dimensional multi-qudit systems and multipartite systems of subsystems with different dimensions. It is proved that these criteria can detect the k -nonseparability ( k is even) of multipartite qudit systems and arbitrary high dimensional multipartite systems of m subsystems with different dimensions. We show that they are more efficient and wider of application range than the previous ones. They provide experimental implementation in detecting entanglement without full quantum state tomography.
Abstract From a practical perspective it is advantageous to develop methods that verify entanglement in quantum states with as few measurements as possible. In this paper we investigate the minimal number of mutually unbiased bases (MUBs) needed to detect bound entanglement in bipartite ( d × d ) -dimensional states, i.e. entangled states that are positive under partial transposition. In particular, we show that a class of entanglement witnesses (EWs) composed of MUBs can detect bound entanglement if the number of measurements is greater than d / 2 + 1 . This is a substantial improvement over other detection methods, requiring significantly fewer resources than either full quantum state tomography or measuring a complete set of d + 1 MUBs. Our approach is based on a partial characterisation of the (non-)decomposability of EWs. We show that non-decomposability is a universal property of MUBs, which holds regardless of the choice of complementary observables, and we find that both the number of measurements and the structure of the witness play an important role in the detection of bound entanglement.
Abstract Mutually unbiased bases (MUBs) provide a standard tool in the verification of quantum states, especially when harnessing a complete set for optimal quantum state tomography. In this work, we investigate the detection of entanglement via inequivalent sets of MUBs, with a particular focus on unextendible MUBs. These are bases for which an additional unbiased basis cannot be constructed and, consequently, are unsuitable for quantum state verification. Here, we show that unextendible MUBs, as well as other inequivalent sets in higher dimensions, can be more effective in the verification of entanglement. Furthermore, we provide an efficient and systematic method to search for inequivalent MUBs and show that such sets occur regularly within the Heisenberg–Weyl MUBs, as the dimension increases. Our findings are particularly useful for experimentalists since they demonstrate that a clever selection of MUBs allows for entanglement detection with fewer measurements.
We investigate whether paradigmatic measurements for quantum state tomography, namely mutually unbiased bases and symmetric informationally complete measurements, can be employed to certify quantum correlations. For this purpose, we identify a simple and noise-robust correlation witness for entanglement detection, steering, and nonlocality that can be evaluated based on the outcome statistics obtained in the tomography experiment. This allows us to perform state tomography on entangled qutrits, a test of Einstein-Podolsky-Rosen steering and a Bell inequality test, all within a single experiment. We also investigate the trade-off between quantum correlations and subsets of tomographically complete measurements as well as the quantification of entanglement in the different scenarios. Finally, we perform a photonics experiment in which we demonstrate quantum correlations under these flexible assumptions, namely with both parties trusted, one party untrusted and both parties untrusted.
Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information processing. Standard methods to detect genuine multipartite entanglement, e.g., entanglement witnesses, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in an experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Bell-like correlations of measurement outcomes collected in the experiment, namely, device independently. However, it is difficult to certify genuine entanglement of practical multipartite states in this way, and even more difficult to quantify it, due to the difficulty in identifying optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this Letter, we explore a general and robust device-independent method that can be applied to various realistic multipartite quantum states in arbitrary finite dimension, while merely relying on bipartite Bell inequalities. Our method allows us both to certify the presence of genuine multipartite entanglement and to quantify it. Several important classes of entangled states are tested with this method, leading to the detection of genuinely entangled states. We also certify genuine multipartite entanglement in weakly entangled Greenberger-Horne-Zeilinger states, showing that the method applies equally well to less standard states.