Certification of high-dimensional entanglement in ultracold atom systems

Niklas Euler, Martin Gärttner

Quantum entanglement has been identified as a crucial concept underlying many intriguing phenomena in condensed matter systems, such as topological phases or many-body localization.Recently, instead of considering mere quantifiers of entanglement such as entanglement entropy, the study of entanglement structure in terms of the entanglement spectrum has shifted to a focus leading to new insights into fractional quantum Hall states and topological insulators, among others.What remains a challenge is the experimental detection of such fine-grained properties of quantum systems.The development of protocols for detecting features of the entanglement spectrum in cold-atom systems, which are one of the leading platforms for quantum simulation, is thus highly desirable and will open up new avenues for experimentally exploring quantum many-body physics.Here, we present a method to bound the width of the entanglement spectrum, or entanglement dimension, of cold atoms in lattice geometries, requiring only measurements in two experimentally accessible bases and utilizing ballistic time-of-flight (TOF) expansion.Building on previous proposals for entanglement certification for photon pairs, we first consider entanglement between two atoms of different atomic species and later generalize to higher numbers of atoms per species and multispecies configurations showing multipartite high-dimensional entanglement.Through numerical simulations, we show that our method is robust against typical experimental noise effects and thus will enable high-dimensional entanglement certification in systems of up to eight atoms using currently available experimental techniques.

Manuel Erhard, Mario Krenn, Anton Zeilinger

Gabriele De Chiara, Anna Sanpera

Ohad Lib, Shuheng Liu, Ronen Shekel +4 more

High-dimensional entangled states offer higher information capacity and stronger resilience to noise compared with two-dimensional systems. However, the large number of modes and sensitivity to random rotations complicate experimental entanglement certification. Here, we experimentally certify three-dimensional entanglement in a five-dimensional two-photon state using 800 Haar-random measurements implemented via a ten-plane programmable light converter. We further demonstrate the robustness of this approach against random rotations, certifying high-dimensional entanglement despite arbitrary phase randomization of the optical modes. This method, which requires no common reference frame between parties, opens the door for high-dimensional entanglement distribution through long-range random links.

Nicolas Laflorencie

Alexandria J. Moore, Andrew M. Weiner

Entanglement -- the coherent correlations between parties in a joint quantum system -- is well-understood and quantifiable in the two-dimensional, two-party case. Higher (>2)-dimensional entangled systems hold promise in extending the capabilities of various quantum information applications. Despite the utility of such systems, methods for quantifying high-dimensional entanglement are more limited and experimentally challenging. We review entanglement certification approaches and the large number of -- often difficult -- measurements required to apply them. We present a novel certification method whose measurement requirements scale linearly with subsystem dimension (scaling with the square-root of the system dimension) and which requires only a single complex measurement. The certification method places a lower-bound on the relative entropy of entanglement of any maximally correlated state thereby certifying system entanglement. A lower bound is also shown for any maximally correlated state in the presence of noise -- the expected experimental case. We discuss experimental realization of all required measurements.