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arXiv:1805.00377 · Physical Review A

Semi-device-independent characterization of multipartite entanglement of states and measurements

Armin Tavakoli, Alastair A. Abbott, Marc-Olivier Renou, Nicolas Gisin, Nicolas Brunner

Published Nov 26, 2018• 30 citations
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Abstract

The semi-device-independent framework allows one to draw conclusions about properties of an unknown quantum system under weak assumptions. Here we present a semi-device-independent scheme for the characterization of multipartite entanglement based around a game played by several isolated parties whose devices are uncharacterized beyond an assumption about the dimension of their Hilbert spaces. Our scheme can simultaneously certify that an $n$-partite high-dimensional quantum state features genuine multipartite entanglement, and that a joint measurement on $n$ subsystems is entangled. Moreover, it provides a lower bound on the number of entangled measurement operators. These tests are strongly robust to noise, and even optimal for certain classes of states and measurements, as we demonstrate with illustrative examples. Notably, our scheme allows the certification of many entangled states admitting a local model, which therefore cannot violate any Bell inequality.

DOI
10.1103/physreva.98.052333

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The semi-device-independent framework allows one to draw conclusions about properties of an unknown quantum system under weak assumptions. Here we present a semi-device-independent scheme for the characterization of multipartite entanglement based around a game played by several isolated parties whose devices are uncharacterized beyond an assumption about the dimension of their Hilbert spaces. Our scheme can simultaneously certify that an nnn-partite high-dimensional quantum state features genuine multipartite entanglement, and that a joint measurement on nnn subsystems is entangled. Moreover, it provides a lower bound on the number of entangled measurement operators. These tests are strongly robust to noise, and even optimal for certain classes of states and measurements, as we demonstrate with illustrative examples. Notably, our scheme allows the certification of many entangled states admitting a local model, which therefore cannot violate any Bell inequality.
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.
Device-independent certification of quantum states enables the characterization of states within a device under minimal physical assumptions. A major problem in this regard is to certify quantum states using minimal resources. Aiming to address this problem, we consider a multipartite quantum steering scenario involving an arbitrary number of parties, of which only one is trusted, meaning that the measurements performed by this party are known. Consequently, the self-testing scheme is almost device-independent. Importantly, all the parties can only perform two measurements each, which is the minimal number of measurements required to observe any form of quantum nonlocality. Then, we propose steering inequalities that are maximally violated by three major classes of genuinely multipartite entangled (GME) states: graph states of arbitrary local dimension, Schmidt states of arbitrary local dimension, and NNN-qubit generalized W states. Using the proposed inequalities, we then provide an almost device-independent certification of the above GME states. Restricting to qubits, we also lift our almost device-independent scheme to device-independent self-testing.
Certifying the entanglement of quantum states with Bell inequalities allows one to guarantee the security of quantum information protocols independently of imperfections in the measuring devices. Here, we present a similar procedure for witnessing entangled measurements, which play a central role in many quantum information tasks. Our procedure is termed semi-device-independent, as it uses uncharacterized quantum preparations of fixed Hilbert space dimension. Using a photonic setup, we experimentally certify an entangled measurement using only measurement statistics. We also apply our techniques to certify unentangled but nevertheless inherently quantum measurements.
We consider the problem of determining whether genuine multipartite entanglement was produced in an experiment, without relying on a characterization of the systems observed or of the measurements performed. We present an n-partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states, but which can be violated by n-partite entangled states, such as Greenberger-Horne-Zeilinger states. In contrast to traditional entanglement witnesses, the violation of this inequality implies that the state is not biseparable independently of the Hilbert space dimension and of the measured operators. Violation of this inequality does not imply, however, genuine multipartite nonlocality. We show more generically how the problem of identifying genuine tripartite entanglement in a device-independent way can be addressed through semidefinite programming.
We propose two semi-device-independent approaches that are able to quantify unknown multipartite quantum entanglement experimentally, where the only information that has to be known beforehand is quantum dimension, and the concept that plays a key role is nondegenerate Bell inequalities. Specifically, using the nondegeneracy of multipartite Bell inequalities, we obtain useful information on the purity of target quantum state. Combined with an estimate of the maximal overlap between the target state and pure product states, and a continuous property of the geometric measure of entanglement, we shall prove the information on purity allows us to give a lower bound for this entanglement measure. In addition, we show that a different combination of the above results also converts to a lower bound for the relative entropy of entanglement. As a demonstration, we apply our approach on 5-partite qubit systems with the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and show that useful lower bounds for the geometric measure of entanglement can be obtained if the Bell expression value is larger than 3.60, and those for the relative entropy of entanglement can be given if the Bell expression value is larger than 3.80, where the Tsirelson bound is 4.
We consider the problem of demonstrating non-Bell-local correlations by performing local measurements in randomly chosen triads, i.e., three mutually unbiased bases, on a multipartite Greenberger-Horne-Zeilinger state. Our main interest lies in investigating the feasibility of using these correlations to certify multipartite entanglement in a device-independent setting. In contrast with previous works, our numerical results up to the eight-partite scenario suggest that if each triad is randomly but uniformly chosen according to the Haar measure, one always (except possibly for a set of measure zero) finds Bell-inequality-violating correlations. In fact, a substantial fraction of these is even sufficient to reveal, in a device-independent manner, various higher-order entanglement. In particular, for the specific cases of three parties and four parties, our results---obtained from semidefinite programming---suggest that these randomly generated correlations always reveal, even in the presence of a non-negligible amount of white noise, the genuine multipartite entanglement possessed by these states. In other words, provided local calibration can be carried out to good precision, a device-independent certification of the genuine multipartite entanglement contained in these states can, in principle, also be carried out in an experimental situation without sharing a global reference frame.