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Consider a Haar random quantum state of depth $d$. Consider any bipartition of this state. According to this paper (page $2$):

Haar-random states on $n$ qudits are nearly maximally entangled across all cuts simultaneously. Random quantum circuits on $L \times L \times \cdots L$ arrays of qudits achieve similar near-maximal entanglement across all possible cuts once the depth is $\Omega(L)$, and before this time, the entanglement often spreads “ballistically."

The comment indicates that the entanglement entropy is near maximal for linear depths. But, how much entanglement entropy is there for logarithmic and constant depths, and what is meant by "ballistic spreading"?

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  • $\begingroup$ The question is a little old, so maybe you already figured it out, but this is saying that the von Neumann entropy of a subsystem becomes maximal when the RQC depth is the width of the graph, which is $O(n^{1/D})$ for a $D$-dimensional array of qubits, so only linear depth in 1D (which is also the design depth for RQCs). But this is for all possible (and arbitrarily large subsystems). How long it takes for the entropy to become maximal depends on the size of the subsystem and how close to maximal you require, for $|A|=O(1)$ $\rho_A$ has near maximal vN entropy after a constant depth. $\endgroup$
    – 4xion
    Dec 15, 2021 at 21:53
  • $\begingroup$ and ballistic here means the entanglement of a region is growing as the lightcone of that region (i.e. almost as fast as possible). the terminology comes from ballistic vs diffusive in condensed matter $\endgroup$
    – 4xion
    Dec 15, 2021 at 21:56

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