Bounding entanglement dimensionality from the covariance matrix

Shuheng Liu1,2,3, Matteo Fadel4, Qiongyi He1,5,6, Marcus Huber2,3, and Giuseppe Vitagliano2,3

1State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria
3Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 1090 Vienna, Austria
4Department of Physics, ETH Zürich, 8093 Zürich, Switzerland
5Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
6Hefei National Laboratory, Hefei 230088, China

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High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC) [1] and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.

High-dimensional entanglement has been identified as an important resource in quantum information processing, but also as a main obstacle for simulating classically a quantum system. In particular, the resource needed to reproduce the correlations in the quantum state can be quantified by the so-called entanglement dimensionality. Because of this, experiments aim at controlling larger and larger quantum systems and prepare them in high-dimensional entangled states. The question arising is then how to detect such entanglement dimensionality from experimental data, for example through specific entanglement witnesses. Most common methods involve very complex measurements, such as fidelities with respect to highly entangled states, which are often challenging and in some cases, like in ensembles of many atoms, completely inaccessible.

To overcome some of these difficulties, we focus here on quantifying entanglement dimensionality through covariances of global observables, which are typically measured in many-body experiments, such as those involving atomic ensembles in highly entangled spin-squeezed states. Concretely, we generalize well-known entanglement criteria based on covariance matrices of local observables and establish analytical bounds for different entanglement dimensionalities, which, when violated, certify what is the minimal entanglement dimensionality present in the system.

To show the practical relevance of our results, we derive criteria that require similar information as the existing methods in literature, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin operators, similar to spin-squeezing inequalities, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems.

As a future outlook, our work also opens interesting research directions and poses further intriguing theoretical questions, such as improving current methods to detect the entanglement dimensionality in multipartite states.

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Cited by

[1] Satoya Imai, Otfried Gühne, and Stefan Nimmrichter, "Work fluctuations and entanglement in quantum batteries", Physical Review A 107 2, 022215 (2023).

[2] Irénée Frérot, Matteo Fadel, and Maciej Lewenstein, "Probing quantum correlations in many-body systems: a review of scalable methods", Reports on Progress in Physics 86 11, 114001 (2023).

[3] Shuheng Liu, Qiongyi He, Marcus Huber, Otfried Gühne, and Giuseppe Vitagliano, "Characterizing Entanglement Dimensionality from Randomized Measurements", PRX Quantum 4 2, 020324 (2023).

[4] Nikolai Wyderka and Andreas Ketterer, "Probing the Geometry of Correlation Matrices with Randomized Measurements", PRX Quantum 4 2, 020325 (2023).

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