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Universal decay of (conditional) mutual information in gapped pure- and mixed-state quantum matter

Jinmin Yi, Kangle Li, Chuan Liu, Zixuan Li, Liujun Zou

Abstract

For spin and fermionic systems in any spatial dimension, we establish that the superpolynomial decay behavior of mutual information and conditional mutual information is a universal property of gapped pure- and mixed-state phases, i.e., all systems in such a phase possess this property if one system in this phase possesses this property. We further demonstrate that the (conditional) mutual information indeed decays superpolynomially in a large class of phases, including chiral phases. As a byproduct, we sharpen the notion of mixed-state phases.

Universal decay of (conditional) mutual information in gapped pure- and mixed-state quantum matter

Abstract

For spin and fermionic systems in any spatial dimension, we establish that the superpolynomial decay behavior of mutual information and conditional mutual information is a universal property of gapped pure- and mixed-state phases, i.e., all systems in such a phase possess this property if one system in this phase possesses this property. We further demonstrate that the (conditional) mutual information indeed decays superpolynomially in a large class of phases, including chiral phases. As a byproduct, we sharpen the notion of mixed-state phases.
Paper Structure (3 theorems, 16 equations, 3 figures)

This paper contains 3 theorems, 16 equations, 3 figures.

Key Result

Theorem 1

Let $H_0$ be a gapped, almost-local Hamiltonian. Suppose that for every (possibly mixed) ground state $\rho$ of $H_0$, any of the following equations holds, then the same equation holds for any $\rho'$ in the same gapped phase as $H_0$. Here $|A|$ and $|B|$ are the sizes of $A$ and $B$, respectively, $\mathrm{dist}(A, C)$ is the distance between $A$ and $C$, and $\mathrm{dist}(A,C)^{-\infty}$ rep

Figures (3)

  • Figure 1: Region $A$ is a contractible region shielded from $C$ by the region $B$. Together, regions $A$, $B$ and $C$ partition the entire lattice. While shown here for a 2D lattice, this partition generalizes to arbitrary dimensions where the region $B$ shields a contractible $A$ from $C$.
  • Figure 2: Decomposition of the adiabatic continuation in 1D, which can be generalized to higher dimensions straightforwardly. (a) Fundamental decomposition adapted from Ref. Haah_HamiltonianEvolution. (b) Decomposition used in this work, obtained by applying (a) four times. Regions $A$, $B$, and $C$ are shown in orange, blue, and green, respectively.
  • Figure 3: Construction of the recovery map for local erasure noise on region $A$. Regions $A$, $B$, and $C$ are shown in orange, blue, and green, respectively. $\mathcal{N}$ denotes the erasure on region $A_+$. The Petz map $\mathcal{E}_{B_-}^P$ acts on $B_-$ with output supported on $B_-CC_+$. The idea is to evolve $\rho'$ back to $\rho$, perform the recovery, and then evolve forward to $\rho'$. On the right-hand side, certain evolutions cancel with their conjugates, yielding $\tilde{\mathcal{E}}_{B}^P$ supported entirely on region $B$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Definition 1