Witnessing Short- and Long-Range Nonstabilizerness via the Information Lattice

Abstract

We study nonstabilizerness on the information lattice, and demonstrate that noninteger local information directly indicates nonstabilizerness. For states with a clear separation of short- and large-scale information, noninteger total information at large scales $Γ$ serves as a witness of long-range nonstabilizerness. We propose a folding procedure to separate the global and edge-to-edge contributions to $Γ$. As an example we show that the ferromagnetic ground state of the spin-1/2 three-state Potts model has long-range nonstabilizerness originating from global correlations, while the paramagnetic ground state has at most short-range nonstabilizerness.

Figures (2)

• Figure 1: Information lattice of (a) $\vert$Néel$\rangle=\ket{0101}$, (b) $\ket{\rm Bell}=\frac{1}{\sqrt{2}}(\ket{0101}+\ket{0011})$, and (c) $\ket{\rm GHZ}=\frac{1}{\sqrt{2}}(\ket{0000}+\ket{1111})$. The right side shows the stabilizer group $\mathcal{G}_{\rm GHZ}$ of the four-qubit GHZ state (excluding the identity operator). (d) Information lattice of a stabilizer state constructed by applying a 2-layer random Clifford circuit $\mathcal{U}_{\mathcal{C}}$ to the ten-qubit GHZ state. (e) Information lattice of a short-range nonstabilizer state constructed by applying a random $T$-doped Clifford circuit $\mathcal{U}_{\mathcal{C}T}$ on the product state $\ket{0}^{\bigotimes 10}$. The circuit $\mathcal{U}_{\mathcal{C}T}$ is composed of three blocks, each one containing $10$ layers of Clifford gates and $5$$T$-gates acting on randomly-selected qubits and between two randomly selected Clifford layers. (f, g) Information per scale $I^\ell$ of (d) and (e) respectively. In (a-e) the sites of the information lattice with integer values of local information are highlighted with a bold black line encircling the site.
• Figure 2: (a) Sketch of the folding procedure used to estimate $\gamma_{\rm global}$ in a chain of length $L=6$. The chain is folded across the gray dashed line at its middle point, resulting in a folded chain of length $L'=3$. The sites in the folded chain are given by the blue ovals, denoted by $n'$, and encompass those sites of the unfolded chain, $n$, belonging to each oval. The dotted red line sketches an edge-to-edge correlation between sites $n=0$ and $n=5$ in the unfolded chain, which becomes completely local to site $n'=0$ in the folded chain. (b) The total information at large scales in the unfolded and folded chain of the symmetric ground state of the spin-$1/2$ three-state Potts model, as a function of the magnetic field $h$ and system size $L$, with $J=1$. The right inset depicts the information lattice of the symmetric ground state at $h=0$ and $L=12$. The left inset shows the information per scale for the same state (blue line) and for the paramagnetic ground state at $h=0.75$ and $L=12$.