Bounding entanglement dimensionality from the covariance matrix

Abstract

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.

25 pages

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Cited by in corpus (12)

• Enhanced Schmidt number criteria based on correlation trace norms
• Semidefinite relaxations for high-dimensional entanglement in the steering scenario
• 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
• Estimating the Schmidt numbers of quantum states via symmetric measurements
• Uncertainty relations between quantum Fisher information and entanglement monotones
• $su(d)$-squeezing and many-body entanglement geometry in finite-dimensional systems
• Quantum null-hypothesis device-independent Schmidt rank witness
• A note on Schmidt-number witnesses based on symmetric measurements
• On the characterization of partially entanglement breaking and annihilating channels
• Certifying entanglement dimensionality by $k$-reduction moments