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arXiv:2004.09101 · Physical Review A

Detection loophole in measurement-device-independent entanglement witnesses

Kornikar Sen, Chirag Srivastava, Shiladitya Mal, Aditi Sen, Ujjwal Sen

Published Mar 15, 2021• 8 citations
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Abstract

There always exists an entanglement witness for every entangled quantum state. Negativity of the expectation value of an entanglement witness operator guarantees entanglement of the corresponding state, given that the measurement devices involved are perfect, i.e., the performed measurements actually constitute the witness operator for the state under consideration. In a realistic situation, there are two possible ways of measurements to drive the process away from the ideal one. First, wrong measurements may be performed, and, second, while the measurement operators are implemented correctly, the detection process may be noisy. Entanglement witnesses are prone to both of these imperfections. The concept of measurement-device-independent entanglement witnesses was introduced to remove the first problem. We analyze the ``detection loophole'' in the context of measurement-device-independent entanglement witnesses, which deal with the second problem of imprecise measurements. We obtain an upper bound, a threshold value, on the entanglement witness function in the measurement-device-independent entanglement witness scenario, below which entanglement is guaranteed for given nonideal detector efficiencies, that can involve both lost events and dark counts. We also present a threshold value on the function in the ``worst-case scenario'' where the threshold of the witness operator for guaranteeing entanglement assumes its maximal negative value.

DOI
10.1103/physreva.103.032415

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There always exists an entanglement witness for every entangled quantum state. Negativity of the expectation value of an entanglement witness operator guarantees entanglement of the corresponding state, given that the measurement devices involved are perfect, i.e., the performed measurements actually constitute the witness operator for the state under consideration. In a realistic situation, there are two possible ways of measurements to drive the process away from the ideal one. First, wrong measurements may be performed, and, second, while the measurement operators are implemented correctly, the detection process may be noisy. Entanglement witnesses are prone to both of these imperfections. The concept of measurement-device-independent entanglement witnesses was introduced to remove the first problem. We analyze the detection loophole'' in the context of measurement-device-independent entanglement witnesses, which deal with the second problem of imprecise measurements. We obtain an upper bound, a threshold value, on the entanglement witness function in the measurement-device-independent entanglement witness scenario, below which entanglement is guaranteed for given nonideal detector efficiencies, that can involve both lost events and dark counts. We also present a threshold value on the function in the worst-case scenario'' where the threshold of the witness operator for guaranteeing entanglement assumes its maximal negative value.
Entanglement, the essential resource in quantum information processing, should be witnessed in many tasks such as quantum computing and quantum communication. The conventional entanglement witness method, relying on an idealized implementation of measurements, could wrongly conclude a separable state to be entangled due to imperfect detections. Inspired by the idea of a time-shift attack, we construct an attack on the conventional entanglement witness process and demonstrate that a separable state can be falsely identified to be entangled. To close such detection loopholes, based on a recently proposed measurement-device-independent entanglement witness method, we design and experimentally demonstrate a measurement-device-independent entanglement witness for a variety of two-qubit states. By the new scheme, we show that an entanglement witness can be realized without detection loopholes.
Entanglement, a critical resource for quantum information processing, needs to be witnessed in many practical scenarios. Theoretically, witnessing entanglement is by measuring a special Hermitian observable, called an entanglement witness (EW), which has non-negative expected outcomes for all separable states but can have negative expectations for certain entangled states. In practice, an EW implementation may suffer from two problems. The first one is reliability. Due to unreliable realization devices, a separable state could be falsely identified as an entangled one. The second problem relates to robustness. A witness may not be optimal for a target state and fail to identify its entanglement. To overcome the reliability problem, we employ a recently proposed measurement-device-independent entanglement witness scheme, in which the correctness of the conclusion is independent of the implemented measurement devices. In order to overcome the robustness problem, we optimize the EW to draw a better conclusion given certain experimental data. With the proposed EW scheme, where only data postprocessing needs to be modified compared to the original measurement-device-independent scheme, one can efficiently take advantage of the measurement results to maximally draw reliable conclusions.
The concept of entanglement witnesses forms a useful technique to detect entanglement in realistic quantum devices. Measurement-device-independent nonlinear entanglement witnesses (MDI-NEWs) are a kind of entanglement witness that eliminates dependence on the correct alignments of measurement devices for guaranteeing the existence of entanglement and also detects more entangled states than its linear counterparts. While this method guarantees entanglement independent of measurement alignments, it is still prone to serving wrong results due to other loopholes. Here, we study the response of MDI-NEWs to two categories of faults occurring in experiments. In the first category, the detection loophole, characterized by lost and additional events of outcomes of measurements, is investigated, and bounds which guarantee entanglement are obtained in terms of the efficiency of measurement being performed. In the second category, we study noise associated with the sets of additional quantum inputs required in MDI-NEW scenarios. In this case, a type of noise is identified which still allows the MDI-NEWs to guarantee entanglement. We also show that MDI-NEWs are less or equally robust in comparison to their linear counterparts under the same noise in additional quantum inputs, although the former group detects a larger volume of entangled states in the noiseless scenario than their linear cousins.
Violation of a Bell inequality certifies that the underlying state must be entangled in a device-independent way, although there exist some entangled states which do not violate such an inequality. However, for every entangled state, it is possible to find a Hermitian operator called an entanglement witness that can detect entanglement through some local measurements in a device-dependent method, but implementation of wrong measurements may lead to fake detection of entanglement. To avoid such difficulties, measurement-device-independent entanglement witness (MDI-EW) based on a semiquantum nonlocal game was proposed, which is not only robust against wrong measurements but also against a specific kind of lossy detectors. We employ here a measurement-device-independent entanglement witness to detect entanglement in a scenario where half of an entangled pair is possessed by a single observer while the other half is with multiple observers performing unsharp measurements, sequentially, independently, and preserving entanglement as much as possible. Interestingly, we find that the numbers of successful observers who can detect entanglement, measurement-device-independently, both with equal and unequal sharpness parameters of the noisy measurements, are greater than that obtained with standard and Bell-inequality-based entanglement detection methods, reflecting its robustness. The entanglement contents of the sequentially shared states are also analyzed. Unlike other scenarios, our investigations also reveal that in this measurement-device-independent situation, states having entanglement in proximity to maximal remain entangled until there are two sequential observers, even if they measure sharply.
Experimental detection of entanglement of an arbitrary state of a given bipartite system is crucial for exploring many areas of quantum information processing. But such a detection should be made in a device-independent way if the preparation process of the state is considered to be faithful, in order to avoid detection of a separable state as an entangled one. The recently developed scheme of detecting bipartite entanglement in a measurement-device-independent way [Phys. Rev. Lett. 110, 060405 (2013)] does require information about the state. Here, by using Auguisiak et al.'s universal entanglement witness scheme for two-qubit states [Phys. Rev. A 77, 030301 (2008)], we provide a universal entanglement detection scheme for two-qubit states in a measurement-device-independent way. We also provide a set of universal witness operators for detecting NPT-ness (negative under partial transpose) of two-qudit states in a measurement-device-independent way. We conjecture that no such universal entanglement witness scheme exists for PPT (positive under partial transpose) entangled states. We also analyze the robustness of some of the experimental schemes---for detecting entanglement in a measurement-device-independent way---under the influence of noise in the inputs (from the referee) as well as in the measurement operator.
Entanglement witnesses form an effective method to locally detect entanglement in the laboratory without having the prior knowledge of the full density matrix. However, separable states can be erroneously indicated as entangled in such detections in the presence of wrong measurements or loss in detectors. Measurement-device-independent entanglement witnesses (MDI-EWs) never detect fake entanglement even under wrong measurements and for a particular kind of lossy detectors. A crucial assumption in the case of faithful detection of entanglement employing MDI-EWs is that the preparation devices producing ``quantum inputs''---which are inputs additional to the quantum state whose entanglement is to be detected---are perfect and there is no noise during their transmission. Here we relax these assumptions and provide a general framework for studying the effect of noise on the quantum inputs, invoking uniform and nonuniform noise models. We derive sufficient conditions on the uniform noisy map for retaining the characteristic of MDI-EWs. We find that in the context of nonuniform and entangling noise, fake entanglement detection is possible even by MDI-EWs. We also investigate various paradigmatic models of local noise and find conditions of revealing entanglement in the class of Werner states.