Persistent homology of quantum entanglement

Structure in quantum entanglement entropy is often leveraged to focus on a small corner of the exponentially large Hilbert space and efficiently parametrize the problem of finding ground states. A typical example is the use of matrix product states for local and gapped Hamiltonians. We study the structure of entanglement entropy using persistent homology, a relatively new method from the field of topological data analysis. The inverse quantum mutual information between pairs of sites is used as a distance metric to form a filtered simplicial complex. Both ground states and excited states of common spin models are analyzed as an example. Furthermore, the effect of homology with different coefficients and boundary conditions is also explored. Beyond these basic examples, we also discuss the promising future applications of this modern computational approach, including its connection to the question of how space-time could emerge from entanglement.

Figure 1

A given state $\psi$ in Hilbert space is mapped to a barcode that describes the topological structure of the entanglement of its subsystems. The proximity parameter $\epsilon$ is computed from the quantum mutual information and the homology groups indicate features in the emergent geometry.

Figure 2

Vietoris-Rips complex of a point cloud with six data points (a) at increasing proximity parameter $\epsilon$ (b)–(e). The barcode (f) shows the persistent homology.

Figure 3

Here we show (a) 0-simplex (point), (b) 1-simplex (line segment), (c) 2-simplex (filled triangle), (d) 3-simplex (filled tetrahedron). Higher-dimensional simplices exist but are not shown here for simplicity. Simplices are glued together to form a simplicial complex and its topological properties are described by simplicial homology. Entangled quantum subsystems $A, B, C$ and $D$ are represented by 0-simplices. A $k$-simplex consists of $k+1$ subsystems.

Figure 4

Example of a simplicial complex created from a quantum system comprising four subsystems: $A, B, C$, and $D$. The entanglement entropy between all subsystems is used to compute the distance $D_{ij}$ [Eq. (11)]. This complex corresponds length scale $\epsilon$, where in this case subsystems $A, B$, and $C$ form a 2-simplex. Subsystem $D$ is not connected to the $ABC$ component, indicating that the distances between $D$ and $A, B$, and $C$ is larger than $\epsilon$. Here, the Betti numbers are ${\beta }_0=2$ (i.e., two connected components: $ABC$ and $D$) and ${\beta }_1=0$ (no one-dimensional holes, since the 1-cycle $ABC$ is also the boundary of the 2-simplex $ABC$).

Figure 5

(a) Half-chain entanglement entropy $S$ as a function of field $h$ in the 1D TFIM. For small fields $h$, the entanglement is constant, $ln\left(2\right)$. At the critical point of $h=1, S$ grows as $ln\left(N\right)$. (b) Quantum mutual information between first site 1 and site $j$ as a function of field $h$ (logarithmically spaced) in the 1D TFIM. The mutual information $M_{1j}$ is largest between nearby sites, indicating the local interactions of the Hamiltonian. Periodic boundary conditions are used and lead to increased $M_{1j}$ for large $j$.

Figure 6

Barcodes for transverse-field Ising chain with $L=10$ sites at (a) $h=0.1$, (b) 1, and (c) 10. The proximity parameter $\epsilon$ refers to the length scale set by the distance matrix $D(i,j)$. The dotted lines and panels above show the simplicial complex for a given $\epsilon$. At the quantum critical point, the spin chain forms a cycle of entanglement with a one-dimensional hole. (d) Lifetime of $H_1$ and $H_0$ bars. The persistence of the 1-cycle at the QCP increases with system size $L$.

Figure 7

Birth and death of the $H_1$ persistence bar at the quantum critical point $h=1$ of the transverse-field Ising chain of varying size $L$. The birth value is set by the distance between neighboring sites and converges to $D_{01}/2ln2\approx 0.744293$. The death value is set by the distance between two maximally separated sites and converges to $D_{0\infty }/2ln2=1$ (see Appendix pp4).

Figure 8

Betti numbers ${\beta }_0$ and ${\beta }_1$ for varying disorder strength $W$. XXZ spin chain with $L=16$ sites, $N_d=3000$ realizations and $k=50$ states.

Figure 9

(a) Maximum Betti number of the first homology group for varying disorder strengths $W$. 3000 disorder realizations. (b) The disorder strength with largest Betti number, i.e., peak in (a), that extrapolates to a critical strength of $W_c\approx 4.79$ (marked by dotted line).

Figure 10

(a) Möbius strip that is triangulated by gluing together 2-simplices. The faces are four unique 1-simplices and two 0-simplices. (b) Simplicial complex of the Möbius strip showing the orientation of the simplices. (c) Simplicial complex of the real projective plane that is constructed by gluing the sides of the Möbius strip together—this can be done in ${\mathbb{R}}^4$ without the surface intersecting itself.

Figure 11

Exact two-site entanglement entropy $S_{0r}/2ln2$ (where $r=|i-j|$) for an infinitely long transverse-field Ising chain with periodic boundary conditions. The dotted lines indicate the minimum $S_{01}$ and the maximum $S_{0\infty }$.

Figure 12

The ground-state magnetization $M=|1/N{\sum }_i{\sigma }_i^x|$ of the transverse-field Ising chain with open boundary conditions (a). Range of the set of zero-homology deaths, where the maximum range coincides with the transition close to $h=1$.