Extracting Entanglement Geometry from Quantum States

Tensor networks impose a notion of geometry on the entanglement of a quantum system. In some cases, this geometry is found to reproduce key properties of holographic dualities, and subsequently much work has focused on using tensor networks as tractable models for holographic dualities. Conventionally, the structure of the network—and hence the geometry—is largely fixed a priori by the choice of the tensor network ansatz. Here, we evade this restriction and describe an unbiased approach that allows us to extract the appropriate geometry from a given quantum state. We develop an algorithm that iteratively finds a unitary circuit that transforms a given quantum state into an unentangled product state. We then analyze the structure of the resulting unitary circuits. In the case of noninteracting, critical systems in one dimension, we recover signatures of scale invariance in the unitary network, and we show that appropriately defined geodesic paths between physical degrees of freedom exhibit known properties of a hyperbolic geometry.

Figure 1

Example of a two-local unitary circuit, where each unitary acts only on the two qubits that are at its ends. The thick red line indicates a geodesic between the fifth and ninth qubit (from the left), following a path through the circuit as given by Fig. 2.

Figure 2

Left panel: labeling of the input and output indices on a unitary operator. Right panel: local graph corresponding to the unitary operator, with weights labeled on the internal edges.

Figure 3

Geodesic length of the $L=500$ Anderson disorder model for different values of the disorder strength $W$, with 200 realizations each. While the physical distance is given in lattice spacings, the geodesic length is in arbitrary units. The inset shows the same data for $W=6.0$, 8.0, 10.0 on a linear scale to highlight the linear dependence $d_g\sim d_p$.

Figure 4

Quench from the ground state of the Anderson disorder model with $L=200$ sites and $W=8$ to the clean case $W=0$. Left panel: geodesic length $d_g$ as a function of physical distance $d_p$. Right panel: entanglement entropy of a contiguous region of $\ell$ sites. Individual lines represent snapshots of the system at times equally spaced between $T=0$ to $T=25.25$ averaged over 200 disorder realizations. The curves in the left panel have been offset by $-15T$. Increasing copper or decreasing blackness indicates times further in the quench.