The hyperlink representation of entanglement and the inclusion-exclusion principle

TL;DR

The paper addresses multipartite entanglement in pure quantum states by elevating the entanglement-link (EL) representation to entanglement hyperlinks (EHLs) via the inclusion–exclusion principle. It defines ${\cal J}_I=\sum_{A\subseteq I}(-1)^{|I|-|A|}S_A$ and develops factorization and reconstruction theorems that yield an exact hyperlink representation of the entanglement entropy: $S_A=\sum_{I\subseteq A}{\cal J}_I$ (bulk) and $S_A=-\tfrac{1}{2}\sum_{I\in A:\bar{A}}{\cal J}_I$ (edge), with a coarse-graining generalization. Numerical studies on 1D free-fermion ground states illustrate factorization and monogamy-like behavior, reveal signs of higher-order EHLs, and demonstrate the efficacy of edge reconstruction for estimating $S_A$ from crossing EHLs. Overall, EHLs provide a powerful framework to diagnose multipartite entanglement and point toward a quantum-geometric perspective beyond area laws, despite computational costs.

Abstract

The entanglement entropy (EE) of any bipartition of a pure state can be approximately expressed as a sum of entanglement links (ELs). In this work, we introduce their exact extension, i.e. the entanglement hyperlinks (EHLs), a type of generalized mutual informations defined through the inclusion-exclusion principle, each of which captures contributions to the multipartite entanglement that are not reducible to lower-order terms. We show that any EHL crossing a factorized partition must vanish, and that the EHLs between any set of blocks can be expressed as a sum of all the EHLs that join all of them. This last result allows us to provide an exact representation of the EE of any block of a pure state, from the sum of the EHLs which cross its boundary. In order to illustrate their rich structure, we discuss some explicit numerical examples using ground states of local Hamiltonians. The EHLs thus provide a remarkable tool to characterize multipartite entanglement in quantum information theory and quantum many-body physics.

Figures (8)

• Figure 1: Illustration of the factorization theorem for EHL. If $S(A)=S(\bar{A})=0$, and $I\in A:\bar{A}$, then we may define $I_1=I\cap A$ and $I_2=I\cap \bar{A}$ and show that ${\cal I}(I_1,I_2)=0$. Then, the factorization theorem implies that ${\cal J}_I=0$.
• Figure 2: Graphical representation of the edge reconstruction of the EE of block $A=\lbrace1,2\rbrace$, Eq. \ref{['eq:edge_recons_i']}. The EE of block $A$ is (minus one half of) the sum of all the EHLs which cross the boundary between $A$ and $\bar{A}$. We can see (a) the two-legged EHLs, (c) and (d) the three-legged EHLs, (d) and (e) the four-legged EHLs, (f) the single five-legged EHL (which is zero by construction).
• Figure 3: Checking the first factorization conjecture, Eq. \ref{['eq:fact1']}. Whenever the minimal entropy of a non-trivial block $S_\text{min}$ is small, the highest-rank EHL $|{\cal J}_\Omega|$ is also small, using chains of $N=4$, 6 an 8 sites, (a) Dimerized chain, for $\delta\in [-1,1]$; (b) Random chain, using 200 samples.
• Figure 4: Checking the second factorization conjecture, Eq. \ref{['eq:fact2']}. We plot ${\cal J}_I$ vs the minimal mutual information within the block, ${\cal I}_\text{min}(I)$ for all blocks of three or more legs obtained from some of the states considered in Fig. \ref{['fig:SminJtot']}: dimerized GS with $\delta=0$, 0.5 and 0.8, a sample from the random chain always using $N=8$.
• Figure 5: Checking the monogamy conjecture, Eq. \ref{['eq:monoconj']}, showing the EHL vs the EE of all blocks extracted from the Gaussian states considered in the previous figures. Notice the exponential envelope, which supports our claim that larger EE give rise to lower values of the EHL. Blocks with $|I|=1$ and 2 are specially marked, since they follow special relations in this case.