15. Assessment in educational settings

Operator Entanglement in Interacting Integrable Quantum Systems: The Case of the Rule 54 Chain

“…This contrasts with the intermediate time behavior, where one faces an "entanglement barrier" [11,12] reminiscent of the generic linear growth of the entanglement entropy of a pure state after a quantum quench [13].In the late 2000s, the physical intuition that it could sometimes be more efficient to simulate the dynamics of operators -e.g. density matrices-rather than the one of pure states spurred another idea [8,[14][15][16][17]: that local observables in Heisenberg picture, O(t) = e iHt Oe −iHt , could also be approximated that way. In an insightful paper, Prosen andŽnidarič [8] observed numerically that there was a crucial distinction to be made between chaotic [18,19] and non-interacting dynamics: the bond dimension necessary for an MPO representation of O(t) was apparently blowing up exponentially with t in the former case and polynomially in the latter.…”

“…In complete analogy with state entanglement, one defines the OE as S(O) ≡ − i λ i ln λ i . The OE was first introduced in the context of quantum information [20] and later connected to MPOsimulability of quantum dynamics [9,11,[14][15][16][17]21]. In the past months, there has been growing interest in the OE, both in condensed matter and in high-energy theory where it connects to quantum chaos, black holes, complexity and models of emergent spacetime [22][23][24][25][26][27].…”

“…Under time evolution the local operator O(t) = e iHt Oe −iHt -or U −t OU tspreads. Like others before us [11,[14][15][16][17]22], we want to understand how S(O(t)) grows with time, for the bipartition A = (−∞, x], B = (x, ∞). In Refs.…”

“…In Refs. [14,15] it was found numerically that the OE grows at most logarithmically with time in systems with underlying noninteracting fermion dynamics (see Ref. [11] for an analytic derivation), while the behavior of OE in chaotic systems seems to be strikingly different, exhibiting linear growth [22].…”