Sequential Generation of Two-dimensional Super-area-law States with Local Parent Hamiltonian

TL;DR

By mapping a $d$-dimensional stochastic surface growth process to a $(d+1)$-dimensional quantum state, the paper constructs a family of two-dimensional, highly entangled states whose entanglement scaling transitions from area-law to volume-law as the deposition probability $p$ crosses the critical point $p_c=1/2$, interpolating between Edwards-Wilkinson and KPZ dynamics. The entanglement structure is captured by the formula $S = \alpha \langle A \rangle + S_{\mathrm{uncolored}}$ with $\alpha = \ln 2$, and the phase diagram shows $S \sim L$ for $p<1/2$, $S \sim L^{5/4}$ at $p=1/2$, and $S \sim L^2$ for $p>1/2$. A sequential generation protocol using local quantum channels and an explicit local, frustration-free parent Hamiltonian is provided, linking classical stochastic dynamics to quantum many-body ground states and suggesting practical routes for quantum simulation and tensor-network representations. The work highlights how well-understood statistical mechanics models can be harnessed to design and analyze exotic quantum states with prescribed entanglement properties and motivates extensions to other dynamics and experimental realizations.

Abstract

We construct examples of highly entangled two-dimensional states by exploiting a correspondence between stochastic processes in $d$ dimensions and quantum states in $d+1$ dimensions. The entanglement structure of these states, which we explicitly calculate, can be tuned between area law, sub-volume law, and volume law. This correspondence also enables a sequential generation protocol: the states can be prepared through a series of unitary transformations acting on an auxiliary system. We also discuss the conditions under which these states have local, frustration-free parent Hamiltonians.

Figures (7)

• Figure 1: Illustration of the deposition-evaporation model. (a) Under the deposition with probability $p$, while the general intent is to add two blocks, the height difference constraint determines the outcome. Specifically, in the middle and right case where adding two blocks violates the height difference constraint, the only valid deposition leaves the surface invariant, as summarized in \ref{['eq:dep']}. (b) Under the evaporation with probability $1-p$, while the general intent is to remove two blocks, the height difference constraint determines the outcome. Specifically, in the middle and right case where removing two blocks violates the height difference constraint, the only valid evaporation leaves the surface invariant, as summarized in \ref{['eq:eva']}. (c) The invalid moves that violate the height difference restriction.
• Figure 2: (a) We place the height fields on the plaquettes and the physical degrees of freedom, spins, on the edges. The horizontal direction labels the spatial direction $i$ of the dynamics, and the vertical direction labels the temporal direction $t$. Spins determine the height field differences. Four spins around a vertex at $(i,t+1)$ define an update from $h_i(t+1)$ to $h_i(t+2)$ while keeping $h_{i\pm1}(t+1)=h_{i\pm1}(t+2)$ unchanged. (b) Examples of spin configurations that correspond to depositing two blocks and evaporating two blocks. (c) Example of a spin configuration with no blocks removed or added. (d) Example of color-matching: The same two red blocks deposited at time $t$ and evaporated at $t'$, then the two vertices corresponding to these two moves should both be labeled as $r$.
• Figure 3: The visualization of entanglement entropy through the cross-section of a time-slice. The distribution of the contour gives $S_\text{uncolored}$ and the color matching under the contour gives $\langle A\rangle$ area scaling.
• Figure 4: The entanglement phase diagram of the state generated by the deposition-evaporation model with color matching.
• Figure 5: (a) The initialized stacks for sequential generation and the corresponding height configuration of the surface. (b) Schematic of sequential generation: Left—sequential application of unitaries $U_n$; right—resulting state aligned with each $U_n$. Spin qubits (blue dots), color qutrits (blue squares), and emitter stacks (red squares labeled $\mathsf{E}/\mathsf{F}$) evolve as follows: red lines trace emitter dynamics, blue lines trace radiated qubits/qutrits. Gray tilted lines denote the lattice geometry of radiated degrees of freedom. At step $n$, $U_n$ acts on the emitter and newly introduced qubits/qutrits, which are then radiated to form a horizontal slice (blue, between dashed lines aligned with $U_n$ and $U_{n+1}$) of the full state. Bottom gray spins represent the fixed initial condition (not part of the generation).