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arXiv:2305.00488 · Physical Review A

Effects of the detection loophole on rival entanglement attestation techniques

Paranjoy Chaki, Kornikar Sen, Ujjwal Sen

Published May 5, 2026• 0 citations
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Abstract

Loopholes present in an experimental set-up can significantly affect the reliability of entanglement detection. We discuss two methods for detection of entanglement: one is by using the positive partial transposition criterion after quantum state tomography and the other by estimating the second and third moments of partial transposition of the quantum state through random classical snapshots. We examine the impact of inaccuracies in these detection methods by considering presence of spurious clicks or suppression of valid clicks in the detectors. By comparing the two methods, we observe that the condition based on partial transposition moments is more robust to missing counts than the positive partial transposition criteria. Moreover, we realize that in the presence of additional counts, none of the criteria misinterpret any separable state as entangled. But in such a scenario, the condition based on the moments can not guarantee any state as entangled, unless the additional event efficiency is about 0.9 or higher.

DOI
10.1103/drnv-m1nh

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Loopholes present in an experimental set-up can significantly affect the reliability of entanglement detection. We discuss two methods for detection of entanglement: one is by using the positive partial transposition criterion after quantum state tomography and the other by estimating the second and third moments of partial transposition of the quantum state through random classical snapshots. We examine the impact of inaccuracies in these detection methods by considering presence of spurious clicks or suppression of valid clicks in the detectors. By comparing the two methods, we observe that the condition based on partial transposition moments is more robust to missing counts than the positive partial transposition criteria. Moreover, we realize that in the presence of additional counts, none of the criteria misinterpret any separable state as entangled. But in such a scenario, the condition based on the moments can not guarantee any state as entangled, unless the additional event efficiency is about 0.9 or higher.
For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.
Abstract We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k -th condition involves comparing moments of the partially transposed density operator up to order k . Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.
Higher-dimensional entanglement is a valuable resource for several quantum information processing tasks, and is often characterized by the Schmidt number and specific classes of entangled states beyond qubit-qubit and qubit-qutrit systems. We propose a criterion to detect higher-dimensional entanglement, focusing on determining the Schmidt number of quantum states and identifying significant classes of positive partial transposition and negative partial transposition entangled states. Our approach relies on evaluating moments of generalized positive maps which can be efficiently simulated in real experiments without the requirement of full-state tomography. We demonstrate the effectiveness of our detection scheme through various illustrative examples. As a direct application, we explore the implications of our moment-based detection schemes in identifying useful quantum channels such as non-Schmidt-number breaking channels and non-entanglement breaking channels. Finally, we present an operational implication of our proposed moment criterion through its manifestation in channel discrimination tasks.
Abstract Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.
We present a set of inequalities for detecting quantum entanglement of $2\ensuremath{\bigotimes}d$ quantum states. For $2\ensuremath{\bigotimes}2$ and $2\ensuremath{\bigotimes}3$ systems, the inequalities give rise to sufficient and necessary separability conditions for both pure and mixed states. For the case of d>3d>3d>3, these inequalities are necessary conditions for separability, which detect all entangled states that are not positive under partial transposition and even some entangled states with positive partial transposition. These inequalities are given by mean values of local observables and present an experimental way of detecting the quantum entanglement of $2\ensuremath{\bigotimes}d$ quantum states and even multiqubit pure states.
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.