Random unitaries, Robustness, and Complexity of Entanglement

J. Odavić, G. Torre, N. Mijić, D. Davidović, F. Franchini, and S. M. Giampaolo

Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia

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Abstract

It is widely accepted that the dynamic of entanglement in presence of a generic circuit can be predicted by the knowledge of the statistical properties of the entanglement spectrum. We tested this assumption by applying a Metropolis-like entanglement cooling algorithm generated by different sets of local gates, on states sharing the same statistic. We employ the ground states of a unique model, namely the one-dimensional Ising chain with a transverse field, but belonging to different macroscopic phases such as the paramagnetic, the magnetically ordered, and the topological frustrated ones. Quite surprisingly, we observe that the entanglement dynamics are strongly dependent not just on the different sets of gates but also on the phase, indicating that different phases can possess different types of entanglement (which we characterize as purely local, GHZ-like, and W-state-like) with different degree of resilience against the cooling process. Our work highlights the fact that the knowledge of the entanglement spectrum alone is not sufficient to determine its dynamics, thereby demonstrating its incompleteness as a characterization tool. Moreover, it shows a subtle interplay between locality and non-local constraints.

The study explored entanglement dynamics in quantum systems subjected to different sets of local gates. While conventional wisdom suggests that you can predict entanglement dynamics based on the statistical properties of the entanglement spectrum, this research found that the behavior of entanglement not only depended on the set of gates but also on the system's phase. Different phases exhibited distinct types of entanglement, and their response to entanglement cooling varied. This suggests that the entanglement spectrum alone cannot fully characterize entanglement dynamics and highlights a complex interplay between locality and non-local constraints in quantum systems.

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Cited by

[1] Raúl Morral-Yepes, Adam Smith, S.L. Sondhi, and Frank Pollmann, "Entanglement Transitions in Unitary Circuit Games", PRX Quantum 5 1, 010309 (2024).

[2] J. Odavić, G. Torre, N. Mijić, D. Davidović, F. Franchini, and S. M. Giampaolo, "Random unitaries, Robustness, and Complexity of Entanglement", Quantum 7, 1115 (2023).

[3] Gianpaolo Torre, Jovan Odavić, Pierre Fromholz, Salvatore Marco Giampaolo, and Fabio Franchini, "Long-range entanglement and topological excitations", arXiv:2310.16091, (2023).

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