Characterizing entanglement dimensionality from randomized measurements

Abstract

We consider the problem of detecting the dimensionality of entanglement with the use of correlations between measurements in randomized directions. First, exploiting the recently derived covariance matrix criterion for the entanglement dimensionality [S. Liu et al., arXiv:2208.04909], we derive an inequality that resembles well-known entanglement criteria, but contains different bounds for the different dimensionalities of entanglement. This criterion is invariant under local changes of $su(d)$ bases and can be used to find regions in the space of moments of randomized correlations, generalizing the results of [S. Imai et al., Phys. Rev. Lett. 126, 150501 (2021)] to the case of entanglement-dimensionality detection. In particular, we find analytical boundary curves for the different entanglement dimensionalities in the space of second- and fourth-order moments of randomized correlations for all dimensions $d_a=d_b=d$ of a bipartite system. We then show how our method works in practice, also considering a finite statistical sample of correlations, and we also show that it can detect more states than other entanglement-dimensionality criteria available in the literature, thus providing a method that is both very powerful and potentially simpler in practical scenarios. We conclude by discussing the partly open problem of the implementation of our method in the multipartite scenario.

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• Entanglement detection
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• Entanglement Certification $-$ From Theory to Experiment
• The structure of multidimensional entanglement in multipartite systems
• Covariance matrices and the separability problem
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Cited by in corpus (22)

• Analysing quantum systems with randomised measurements
• Probing the geometry of correlation matrices with randomized measurements
• Bounding entanglement dimensionality from the covariance matrix
• Collective randomized measurements in quantum information processing
• Analyzing quantum entanglement with the Schmidt decomposition in operator space
• Enhanced Schmidt number criteria based on correlation trace norms
• Detecting high-dimensional entanglement in cold-atom quantum simulators
• Higher-dimensional entanglement detection and quantum channel characterization using moments of generalized positive maps
• Schmidt number criterion via general symmetric informationally complete measurements
• Experimental certification of high-dimensional entanglement with randomized measurements
• Bound entangled Bell diagonal states of unequal local dimensions, and their witnesses
• Estimating the Schmidt numbers of quantum states via symmetric measurements
• Quantifying subspace entanglement with geometric measures
• Real randomized measurements for analyzing properties of quantum states
• Quantum null-hypothesis device-independent Schmidt rank witness
• Uncertainty relations between quantum Fisher information and entanglement monotones
• Measurement-device-independent Schmidt number certification of all entangled states
• Detecting high-dimensional entanglement by randomized product projections
• A note on Schmidt-number witnesses based on symmetric measurements
• Classification of quantum states of light using random measurements through a multimode fiber
• On the characterization of partially entanglement breaking and annihilating channels
• Certifying entanglement dimensionality by $k$-reduction moments