Engineering entanglement geometry via spacetime-modulated measurements

Abstract

We introduce a general approach to realize quantum states with holographic entanglement structure via monitored dynamics. Starting from random unitary circuits in $1+1$ dimensions, we introduce measurements with a spatiotemporally-modulated density. Exploiting the known critical properties of the measurement-induced entanglement transition, this allows us to engineer arbitrary geometries for the bulk space (with a fixed topology). These geometries in turn control the entanglement structure of the boundary (output) state. We demonstrate our approach by giving concrete protocols for two geometries of interest in two dimensions: the hyperbolic half-plane and a spatial section of the BTZ black hole. We numerically verify signatures of the underlying entanglement geometry, including a direct imaging of entanglement wedges by using locally-entangled reference qubits. Our results provide a concrete platform for realizing geometric entanglement structures on near-term quantum simulators.

Figures (5)

• Figure 1: Visualizing the experimental setup and geodesics. Subfigure (a) shows a dual unitary circuit with random measurements (red) as a function of spacetime and an minimal cut (turquoise). Notice that boundary conditions are important for the BTZ case, but not for the AdS case in which, in principle, has an infinitely deep circuit. In subfigure (b) we show how a large circuit approximates a continuous spacetime with a geodesic. Subfigures (c) and (d) show entanglement geodesics for AdS and a constant-time slice of BTZ respectively, corresponding to states on the boundary in the Poincaré disk model. Notice that the entanglement geodesics vary continuously with size of the subregion in the AdS case while they jump in the BTZ case due to the presence of a horizon.
• Figure 2: Analyzing the relationship between entanglement entropies and geodesic lengths. Subfigures (a) and (c) show the entanglement entropy of a single interval of varying size on the boundary, such as those shown in \ref{['fig:experimental_picture']}. Subfigure (a) shows the entanglement entropy of a region on the boundary of hyperbolic space, while subfigure (c) shows the entanglement entropy of a region on the boundary of a space containing a BTZ black hole with a product state (blue, dotted) or a maximal volume-law state (pink, solid) placed on the horizon. In subfigure (b) we depict the mutual information between two regions of size $\frac{\pi}{4}$ as a function of their separation. No free parameters are needed to establish the mutual information model, as the entropy per unit length is derived from the fit in subfigure (a).
• Figure 3: Imaging the entanglement wedge of two intervals in the BTZ metric. Color plots show mutual information $I(R:AB)$ between a reference qubit $R$ entangled at space-time location $(x,t)$ and the union of two intervals $A$, $B$. The intervals contain $102$ qubits each and are separated by (a) $\delta x = 16$, (b) $\delta x = 32$, (c) $\delta x = 64$ qubits. Orange lines denote contours with $I = 0.75$, clearly showing a transition in the structure of the entanglement wedge as a function of separation $\delta x$. The system size is $L = 512$ and we set $r_h = l = 0.5$, giving $T=128$. Data averaged over $4\times 10^3$ realizations of Clifford circuits.
• Figure S1: Entanglement phase transition in brickwork circuits of dual-unitary Clifford gates and single-qubit measurements. (a) Tripartite mutual information $I_3(A:B:C)$ between contiguous segments of length $L/4$ as a function of measurement density $\rho$ for different system sizes $L$. Data averaged over between $250$ and $10^4$ realizations (depending on $L$) and over time steps $2L < t < 4L$. A finite-size crossing is visible near $\rho = \rho_c \simeq 0.2048$ (vertical dashed line). (b) Scaling collapse of the data as a function of $(\rho - \rho_c) L^{1/\nu}$, with correlation length critical exponent $\nu = 1.30$.
• Figure S2: Logarithmically-entangled states from the AdS metric. (a) Measurement density as a function of time: $\rho(t) = \rho_c[1-(1+|t|/l)^{-1/\nu}]$, for different values of the parameter $l$ which plays the role of AdS radius. The dashed horizontal line indicates $\rho_c = 0.2050$. Inset shows $\rho_c - \rho(t)$ in semilogarithmic scale. (b) Entropy $S_A$ of an interval $A$ as a function of the interval length $|A|$ for several values of $l$. Data from stabilizer simulations of a system of $L = 2048$ qubits, averaged over $> 2000$ circuit realizations. (c) Same data (for $|A|\leq L/2$) in semilogarithmic scale, against fits to $\alpha \log |A| + {\sf const.}$ (dashed lines). (d) Values of the entropy coefficient $\alpha$, extracted from fits to the data in (c), vs AdS radius $l$. We find $\alpha = al+b$ (dashed line). (e) Tripartite mutual information $I_3$ (for consecutive intervals of length $L/4$) vs AdS radius $l$.