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arXiv:1903.09533 · Physical Review A

Optimization of device-independent witnesses of entanglement depth from two-body correlators

Jordi Tura, Albert Aloy, F. Baccari, Antonio Acín, Maciej Lewenstein, Remigiusz Augusiak

Published Sep 5, 2019• 31 citations
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Abstract

In a recent work [A. Aloy et al., Phys. Rev. Lett. 123, 100507 (2019)] we have considered the characterization of entanglement depth, from a device-independent perspective, in a quantum many-body system. We have shown that the inequalities introduced by J. Tura et al. [Science 344, 1256 (2014)] can be used to obtain device-independent witnesses of entanglement depth (DIWEDs) and that they enjoy two key properties that allow one to compute their $k$-producibility bounds more efficiently for larger system sizes, as well as yielding experimentally friendlier device-independent witnesses of entanglement depth: they involve at most two-body correlators and they are permutationally invariant. While the main aim of our previous work was to illustrate the main ideas and applicability of the method, here we outline the details and complement its findings with detailed analysis and further case studies. Specifically, we consider the problem of finding the $k$-producible bounds of such DIWEDs under different assumptions. Not surprisingly, with the weakest assumptions, we can compute $k$-producible bounds only for a relatively small number of parties; however, we can still learn interesting features from these solutions that motivate the search on larger systems under the assumption that these features persist. This allows us to tackle the case where the system size eventually reaches the thermodynamic limit.

DOI
10.1103/physreva.100.032307

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In a recent work [A. Aloy et al., Phys. Rev. Lett. 123, 100507 (2019)] we have considered the characterization of entanglement depth, from a device-independent perspective, in a quantum many-body system. We have shown that the inequalities introduced by J. Tura et al. [Science 344, 1256 (2014)] can be used to obtain device-independent witnesses of entanglement depth (DIWEDs) and that they enjoy two key properties that allow one to compute their kkk-producibility bounds more efficiently for larger system sizes, as well as yielding experimentally friendlier device-independent witnesses of entanglement depth: they involve at most two-body correlators and they are permutationally invariant. While the main aim of our previous work was to illustrate the main ideas and applicability of the method, here we outline the details and complement its findings with detailed analysis and further case studies. Specifically, we consider the problem of finding the kkk-producible bounds of such DIWEDs under different assumptions. Not surprisingly, with the weakest assumptions, we can compute kkk-producible bounds only for a relatively small number of parties; however, we can still learn interesting features from these solutions that motivate the search on larger systems under the assumption that these features persist. This allows us to tackle the case where the system size eventually reaches the thermodynamic limit.
We consider the characterization of entanglement depth in a quantum many-body system from the device-independent perspective; that is, we aim at certifying how many particles are genuinely entangled without relying on assumptions on the system itself nor on the measurements performed. We obtain device-independent witnesses of entanglement depth (DIWEDs) using the Bell inequalities introduced in [J. Tura et al., Science 344, 1256 (2014)SCIEAS0036-807510.1126/science.1247715] and compute their k-producibility bounds. To this end, we exploit two complementary methods: first, a variational one, yielding a possibly optimal k-producible state; second, a certificate of optimality via a semidefinite program, based on a relaxation of the quantum marginal problem. Numerical results suggest a clear pattern on k-producible bounds for large system sizes, which we then tackle analytically in the thermodynamic limit. Contrary to existing DIWEDs, the ones we present here can be effectively measured by accessing only collective measurements and second moments thereof. These technical requirements are met in current experiments, which have already been performed in the context of detecting Bell correlations in quantum many-body systems of 5×10^{2}-5×10^{5} atoms.
Nonlocality detection in multipartite quantum systems is of great interest. The most popular tool to detect nonlocality in quantum systems are Bell inequalities. Most of the provided constructions of multipartite Bell inequalities involve correlations between all parties which quickly becomes computationally intractable and hard to test experimentally in many-body quantum systems. Recently, J. Tura and collaborators have shown in [Science 344.6189 (2014): 1256-1258] that detection of nonlocality in multipartite systems is possible with Bell inequalities involving only one- and two- body correlation functions. However, it is uncertain how efficient these new inequalities are. One of the objectives of the present work is to address this question by numerical means. The other objective is to show that these inequalities can also serve as as device independent witnesses of different forms of entanglement such as genuine multipartite entanglement.
We address the question of assessing the number of particles sharing genuinely nonlocal correlations in a multipartite system. While the interest in multipartite nonlocality has grown in recent years, its existence in large quantum systems is difficult to confirm experimentally. This is mostly due to the inadequacy of standard multipartite Bell inequalities to many-body systems: Such inequalities usually rely on expectation values involving many parties, usually all, and require individual addressing of each party. In addition, known Bell inequalities for genuine nonlocality are composed of a number of expectation values that scales exponentially with the number of observers, which makes such inequalities impractical from the experimental point of view. In a recent work [Tura et al., Science 344, 1256 (2014)], it was shown that it is possible to detect nonlocality in multipartite systems using Bell inequalities with only two-body correlators. This opened the way for the detection of Bell correlations with trusted collective measurements through Bell correlation witnesses [Schmied et al., Science 352, 441 (2016)]. These witnesses were recently tested experimentally in many-body systems such as Bose-Einstein condensate or thermal ensembles, hence demonstrating the presence of Bell correlations with assumptions on the statistics. Here, we address the problem of detecting nonlocality depth, a notion that quantifies the number of particles sharing nonlocal correlation in a multipartite system. We introduce a general framework allowing us to derive Bell-like inequalities for nonlocality depth from symmetric two-body correlators. We characterize all such Bell-like inequalities for a finite number of parties and we show that they reveal Bell correlation depth $k\ensuremath{\le}6$ in arbitrarily large systems. We then show how Bell correlation depth can be estimated using quantities that are within reach in current experiments. On one hand, we use the standard multipartite Bell inequalities such the Mermin and Svetlichny ones to derive Bell correlations witnesses of any depth that involves only two collective measurements, one of which being the parity measurement. On the other hand, we show that our two-body Bell inequalities can be turned into witnesses of depth $k\ensuremath{\le}6$ that require measuring total spin components in certain directions. Interestingly, such a witness is violated by existing data from an ensemble of 480 atoms.
We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, mmm-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.
We present a general method to quantify both bipartite and multipartite entanglement in a device-independent manner, meaning that we put a lower bound on the amount of entanglement present in a system based on the observed data only but independent of any quantum description of the employed devices. Some of the bounds we obtain, such as for the Clauser-Horne-Shimony-Holt Bell inequality or the Svetlichny inequality, are shown to be tight. Besides, device-independent entanglement quantification can serve as a basis for numerous tasks. We show in particular that our method provides a rigorous way to construct dimension witnesses, gives new insights into the question whether bound entangled states can violate a Bell inequality, and can be used to construct device-independent entanglement witnesses involving an arbitrary number of parties.
In a recent paper, Bancal et al. [Phys. Rev. Lett. 106, 250404 (2011)] put forward the concept of device-independent witnesses of genuine multipartite entanglement. These witnesses are capable of verifying genuine multipartite entanglement produced in a laboratory without resorting to any knowledge of the dimension of the state space or of the specific form of the measurement operators. As a by-product they found a multiparty three-setting Bell inequality which makes it possible to detect genuine nnn-partite entanglement in a noisy nnn-qubit Greenberger-Horne-Zeilinger (GHZ) state for visibilities as low as 2/32/32/3 in a device-independent way. In this paper, we generalize this inequality to an arbitrary number of settings, demonstrating a threshold visibility of $2/\ensuremath{\pi}~0.6366$ for number of settings going to infinity. We also present a pseudotelepathy Bell inequality achieving the same threshold value. We argue that our device-independent witnesses are optimal in the sense that for nnn odd the above value cannot be beaten with nnn-party-correlation Bell inequalities.