We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)].
We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)SCIEAS0036-807510.1126/science.aau4963].
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.
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.
The
p3-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of entanglement criteria based on the PT moments
pk=Tr
[(ρΓ)k], where
ρΓ denotes the partially transposed density matrix of a quantum state
ρ. While most of these generalizations are strictly more powerful than the
p3-PPT criterion, their
m-th level versions usually rely on the availability of
pk for all moment orders
k≤m. Here, we show that one can alternatively compare any three PT moments of orders
k<l<m, which can significantly reduce experimental overheads. More precisely, we show that any state satisfying $p_l>p_k^xp_m^{1-x} $ must be entangled, where
x=(m−l)/(m−k). Using the example of locally depolarized GHZ states, we identify the most promising versions of these three-moment criteria and compare their performance with a broad range of entanglement criteria. In the case of globally depolarized stabilizer states, we prove that having access to
pk for
k≤5 is sufficient to reproduce the full PPT criterion. More generally, we show that the Stieltjes-
m criterion is as powerful as the PPT criterion whenever
ρΓ has no more than
(m+1)/2 distinct eigenvalues. Finally, we introduce a notion of quantum weight enumerators that capture the decay of
pk under local white noise for arbitrary quantum states and illustrate this concept for an AME state. Our results contribute to the growing body of literature on higher-moment PPT relaxations and modern applications of weight enumerators in quantum error correction and information theory.
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.