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The nonequilibrium physics of many-body quantum systems harbors various unconventional phenomena. In this Letter, we experimentally investigate one of the most puzzling of these phenomena-the quantum Mpemba effect, where a tilted ferromagnet restores its symmetry more rapidly when it is farther from the symmetric state compared to when it is closer. We present the first experimental evidence of the occurrence of this effect in a trapped-ion quantum simulator. The symmetry breaking and restoration are monitored through entanglement asymmetry, probed via randomized measurements, and postprocessed using the classical shadows technique. Our findings are further substantiated by measuring the Frobenius distance between the experimental state and the stationary thermal symmetric theoretical state, offering direct evidence of subsystem thermalization.

Entanglement is a distinguishing feature of quantum many-body systems, and uncovering the entanglement structure for large particle numbers in quantum simulation experiments is a fundamental challenge in quantum information science 1 . Here we perform experimental investigations of entanglement on the basis of the entanglement Hamiltonian (EH) 2 as an effective description of the reduced density operator for large subsystems. We prepare ground and excited states of a one-dimensional XXZ Heisenberg chain on a 51-ion programmable quantum simulator 3 and perform sample-efficient 'learning' of the EH for subsystems of up to 20 lattice sites 4 . Our experiments provide compelling evidence for a local structure of the EH. To our knowledge, this observation marks the first instance of confirming the fundamental predictions of quantum field theory by Bisognano and Wichmann 5,6 , adapted to lattice models that represent correlated quantum matter. The reduced state takes the form of a Gibbs ensemble, with a spatially varying temperature profile as a signature of entanglement 2 . Our results also show the transition from area- to volume-law scaling 7 of von Neumann entanglement entropies from ground to excited states. As we venture towards achieving quantum advantage, we anticipate that our findings and methods have wide-ranging applicability to revealing and understanding entanglement in many-body problems with local interactions including higher spatial dimensions., (© 2023. The Author(s), under exclusive licence to Springer Nature Limited.)

The control over quantum states in atomic systems has led to the most precise optical atomic clocks so far 1-3 . Their sensitivity is bounded at present by the standard quantum limit, a fundamental floor set by quantum mechanics for uncorrelated particles, which can-nevertheless-be overcome when operated with entangled particles. Yet demonstrating a quantum advantage in real-world sensors is extremely challenging. Here we illustrate a pathway for harnessing large-scale entanglement in an optical transition using 1D chains of up to 51 ions with interactions that decay as a power-law function of the ion separation. We show that our sensor can emulate many features of the one-axis-twisting (OAT) model, an iconic, fully connected model known to generate scalable squeezing 4 and Greenberger-Horne-Zeilinger-like states 5-8 . The collective nature of the state manifests itself in the preservation of the total transverse magnetization, the reduced growth of the structure factor, that is, spin-wave excitations (SWE), at finite momenta, the generation of spin squeezing comparable with OAT (a Wineland parameter 9,10 of -3.9 ± 0.3 dB for only N = 12 ions) and the development of non-Gaussian states in the form of multi-headed cat states in the Q-distribution. We demonstrate the metrological utility of the states in a Ramsey-type interferometer, in which we reduce the measurement uncertainty by -3.2 ± 0.5 dB below the standard quantum limit for N = 51 ions., (© 2023. The Author(s), under exclusive licence to Springer Nature Limited.)